PRDPRVDAQPhysical Review DPhys. Rev. D2470-00102470-0029American Physical Society10.1103/PhysRevD.108.L011103LETTERSObservation of sizeable ω contribution to χc1(3872)→π+π-J/ψ decaysOBSERVATION OF SIZEABLE ω CONTRIBUTION TO …R. 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Collaboration)Centro Brasileiro de Pesquisas Físicas (CBPF), Rio de Janeiro, BrazilUniversidade Federal do Rio de Janeiro (UFRJ), Rio de Janeiro, BrazilCenter for High Energy Physics, Tsinghua University, Beijing, ChinaInstitute Of High Energy Physics (IHEP), Beijing, ChinaSchool of Physics State Key Laboratory of Nuclear Physics and Technology, Peking University, Beijing, ChinaUniversity of Chinese Academy of Sciences, Beijing, ChinaInstitute of Particle Physics, Central China Normal University, Wuhan, Hubei, ChinaUniversité Savoie Mont Blanc, CNRS, IN2P3-LAPP, Annecy, FranceUniversité Clermont Auvergne, CNRS/IN2P3, LPC, Clermont-Ferrand, FranceAix Marseille Univ, CNRS/IN2P3, CPPM, Marseille, FranceUniversité Paris-Saclay, CNRS/IN2P3, IJCLab, Orsay, FranceLaboratoire Leprince-Ringuet, CNRS/IN2P3, Ecole Polytechnique, Institut Polytechnique de Paris, Palaiseau, FranceLPNHE, Sorbonne Université, Paris Diderot Sorbonne Paris Cité, CNRS/IN2P3, Paris, FranceI. Physikalisches Institut, RWTH Aachen University, Aachen, GermanyFakultät Physik, Technische Universität Dortmund, Dortmund, GermanyMax-Planck-Institut für Kernphysik (MPIK), Heidelberg, GermanyPhysikalisches Institut, Ruprecht-Karls-Universität Heidelberg, Heidelberg, GermanySchool of Physics, University College Dublin, Dublin, IrelandINFN Sezione di Bari, Bari, ItalyINFN Sezione di Bologna, Bologna, ItalyINFN Sezione di Ferrara, Ferrara, ItalyINFN Sezione di Firenze, Firenze, ItalyINFN Laboratori Nazionali di Frascati, Frascati, ItalyINFN Sezione di Genova, Genova, ItalyINFN Sezione di Milano, Milano, ItalyINFN Sezione di Milano-Bicocca, Milano, ItalyINFN Sezione di Cagliari, Monserrato, ItalyUniversità degli Studi di Padova, Università e INFN, Padova, Padova, ItalyINFN Sezione di Pisa, Pisa, ItalyINFN Sezione di Roma La Sapienza, Roma, ItalyINFN Sezione di Roma Tor Vergata, Roma, ItalyNikhef National Institute for Subatomic Physics, Amsterdam, NetherlandsNikhef National Institute for Subatomic Physics and VU University Amsterdam, Amsterdam, NetherlandsAGH—University of Science and Technology, Faculty of Physics and Applied Computer Science, Kraków, PolandHenryk Niewodniczanski Institute of Nuclear Physics Polish Academy of Sciences, Kraków, PolandNational Center for Nuclear Research (NCBJ), Warsaw, PolandHoria Hulubei National Institute of Physics and Nuclear Engineering, Bucharest-Magurele, RomaniaAffiliated with an institute covered by a cooperation agreement with CERNICCUB, Universitat de Barcelona, Barcelona, SpainInstituto Galego de Física de Altas Enerxías (IGFAE), Universidade de Santiago de Compostela, Santiago de Compostela, SpainInstituto de Fisica Corpuscular, Centro Mixto Universidad de Valencia—CSIC, Valencia, SpainEuropean Organization for Nuclear Research (CERN), Geneva, SwitzerlandInstitute of Physics, Ecole Polytechnique Fédérale de Lausanne (EPFL), Lausanne, SwitzerlandPhysik-Institut, Universität Zürich, Zürich, SwitzerlandNSC Kharkiv Institute of Physics and Technology (NSC KIPT), Kharkiv, UkraineInstitute for Nuclear Research of the National Academy of Sciences (KINR), Kyiv, UkraineUniversity of Birmingham, Birmingham, United KingdomH.H. Wills Physics Laboratory, University of Bristol, Bristol, United KingdomCavendish Laboratory, University of Cambridge, Cambridge, United KingdomDepartment of Physics, University of Warwick, Coventry, United KingdomSTFC Rutherford Appleton Laboratory, Didcot, United KingdomSchool of Physics and Astronomy, University of Edinburgh, Edinburgh, United KingdomSchool of Physics and Astronomy, University of Glasgow, Glasgow, United KingdomOliver Lodge Laboratory, University of Liverpool, Liverpool, United KingdomImperial College London, London, United KingdomDepartment of Physics and Astronomy, University of Manchester, Manchester, United KingdomDepartment of Physics, University of Oxford, Oxford, United KingdomMassachusetts Institute of Technology, Cambridge, Massachusetts, USAUniversity of Cincinnati, Cincinnati, Ohio, USAUniversity of Maryland, College Park, Maryland, USALos Alamos National Laboratory (LANL), Los Alamos, New Mexico, USASyracuse University, Syracuse, New York, USASchool of Physics and Astronomy, Monash University, Melbourne, Australia (associated with Department of Physics, University of Warwick, Coventry, United Kingdom)Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), Rio de Janeiro, Brazil (associated with Universidade Federal do Rio de Janeiro (UFRJ), Rio de Janeiro, Brazil)Physics and Micro Electronic College, Hunan University, Changsha City, China (associated with Institute of Particle Physics, Central China Normal University, Wuhan, Hubei, China)Guangdong Provincial Key Laboratory of Nuclear Science, Guangdong-Hong Kong Joint Laboratory of Quantum Matter, Institute of Quantum Matter, South China Normal University, Guangzhou, China (associated with Center for High Energy Physics, Tsinghua University, Beijing, China)School of Physics and Technology, Wuhan University, Wuhan, China (associated with Center for High Energy Physics, Tsinghua University, Beijing, China)Departamento de Fisica, Universidad Nacional de Colombia, Bogota, Colombia (associated with LPNHE, Sorbonne Université, Paris Diderot Sorbonne Paris Cité, CNRS/IN2P3, Paris, France)Universität Bonn—Helmholtz-Institut für Strahlen und Kernphysik, Bonn, Germany (associated with Physikalisches Institut, Ruprecht-Karls-Universität Heidelberg, Heidelberg, Germany)Institut für Physik, Universität Rostock, Rostock, Germany (associated with Physikalisches Institut, Ruprecht-Karls-Universität Heidelberg, Heidelberg, Germany)Eotvos Lorand University, Budapest, Hungary (associated with European Organization for Nuclear Research (CERN), Geneva, Switzerland)INFN Sezione di Perugia, Perugia, Italy (associated with INFN Sezione di Ferrara, Ferrara, Italy)Van Swinderen Institute, University of Groningen, Groningen, Netherlands (associated with Nikhef National Institute for Subatomic Physics, Amsterdam, Netherlands)Universiteit Maastricht, Maastricht, Netherlands (associated with Nikhef National Institute for Subatomic Physics, Amsterdam, Netherlands)DS4DS, La Salle, Universitat Ramon Llull, Barcelona, Spain (associated with ICCUB, Universitat de Barcelona, Barcelona, Spain)Department of Physics and Astronomy, Uppsala University, Uppsala, Sweden (associated with School of Physics and Astronomy, University of Glasgow, Glasgow, United Kingdom)University of Michigan, Ann Arbor, Michigan, USA (associated with Syracuse University, Syracuse, New York, USA)*

Full author list given at the end of the article.

Deceased.

Also at Università di Firenze, Firenze, Italy.

Also at Scuola Normale Superiore, Pisa, Italy.

Also at Università di Ferrara, Ferrara, Italy.

Also at Università di Milano Bicocca, Milano, Italy.

Also at Università di Bologna, Bologna, Italy.

Also at Università di Genova, Genova, Italy.

Also at Università di Bari, Bari, Italy.

Also at Universidad Nacional Autónoma de Honduras, Tegucigalpa, Honduras.

Also at Università di Cagliari, Cagliari, Italy.

Also at Departamento de Física Interdisciplinar, Universidad Nacional de Educación a Distancia (UNED), Madrid, Spain.

Also at Università di Roma Tor Vergata, Roma, Italy.

Also at Universidade Federal do Triângulo Mineiro (UFTM), Uberaba-MG, Brazil.

Also at Hangzhou Institute for Advanced Study, UCAS, Hangzhou, China.

Also at Università di Siena, Siena, Italy.

Also at Università degli Studi di Milano, Milano, Italy.

Also at Central South U., Changsha, China.

Also at Università di Padova, Padova, Italy.

Also at Università di Perugia, Perugia, Italy.

Also at Excellence Cluster ORIGINS, Munich, Germany.

Also at Dipartimento MIFT, Universita degli Studi di Messina and INFN Sezione di Catania, Italy, Messina and Catania, Italy.

Also at Università di Pisa, Pisa, Italy.

Also at Università della Basilicata, Potenza, Italy.

Also at Department of Physics, Indiana University, Bloomington, Indiana, USA.

Also at Università di Urbino, Urbino, Italy.

27July20231July20231081L01110326April202217May2023© 2023 CERN, for the LHCb Collaboration2023CERNPublished by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Funded by SCOAP3.

Resonant structures in the dipion mass spectrum from χc1(3872)→π+π-J/ψ decays, produced via B+→K+χc1(3872) decays, are analyzed using proton-proton collision data collected by the LHCb experiment, corresponding to an integrated luminosity of 9  fb-1. A sizeable contribution from the isospin conserving χc1(3872)→ωJ/ψ decay is established for the first time, (21.4±2.3±2.0)%, with a significance of more than 7.1σ. The amplitude of isospin violating decay, χc1(3872)→ρ0J/ψ, relative to isospin conserving decay, χc1(3872)→ωJ/ψ, is properly determined, and it is a factor of 6 larger than expected for a pure charmonium state.

CERN10.13039/100012470Coordenação de Aperfeiçoamento de Pessoal de Nível Superior10.13039/501100002322Conselho Nacional de Desenvolvimento Científico e Tecnológico10.13039/501100003593Fundação Carlos Chagas Filho de Amparo à Pesquisa do Estado do Rio de Janeiro10.13039/501100004586Financiadora de Estudos e Projetos10.13039/501100004809Ministry of Science and Technology of the People’s Republic of China10.13039/501100002855National Natural Science Foundation of China10.13039/501100001809Centre National de la Recherche Scientifique10.13039/501100004794Institut National de Physique Nucléaire et de Physique des Particules10.13039/501100012441Bundesministerium für Bildung und Forschung10.13039/501100002347Deutsche Forschungsgemeinschaft10.13039/501100001659Max-Planck-Gesellschaft10.13039/501100004189Instituto Nazionale di Fisica Nucleare10.13039/501100004007Nederlandse Organisatie voor Wetenschappelijk Onderzoek10.13039/501100003246Ministerstwo Edukacji i Nauki10.13039/501100004569Narodowe Centrum Nauki10.13039/501100004281Ministerul Educaţiei Naţionale10.13039/501100003766Institutul de Fizică Atomică10.13039/501100019278Ministerio de Ciencia e Innovación10.13039/501100004837Schweizerischer Nationalfonds zur Förderung der Wissenschaftlichen Forschung10.13039/501100001711SERNational Academy of Sciences of Ukraine10.13039/501100004742Science and Technology Facilities Council10.13039/501100000271Nuclear Physics10.13039/100006209National Science Foundation10.13039/100000001Australian Research Council10.13039/501100000923Australian Research Data Commons10.13039/501100020840MincienciasAlexander von Humboldt-Stiftung10.13039/100005156EPLANETHORIZON EUROPE Marie Sklodowska-Curie Actions10.13039/100018694European Research Council10.13039/501100000781A*MIDEXAgence Nationale de la Recherche10.13039/501100001665IPhULabex P2IORégion Auvergne-Rhône-Alpes10.13039/501100010115Key Research Program of Frontier Science, Chinese Academy of Sciences10.13039/501100018527CAS PIFICAS Center for Excellence in Particle Physics10.13039/501100015710Fundamental Research Funds for the Central Universities10.13039/501100012226Guangzhou Municipal Science and Technology Project10.13039/501100010256Generalitat Valenciana10.13039/501100003359Xunta de Galicia10.13039/501100010801Generalitat de Catalunya10.13039/501100002809Prog. Atracción Talento, CMSRCLeverhulme Trust10.13039/501100000275Royal Society10.13039/501100000288UK Research and Innovation10.13039/100014013

After the discovery of the χc1(3872) state in decays through the π+π-J/ψ channel [1], the ρ0J/ψ process was suggested to explain the observed π+π- mass (mπ+π-) distribution peaking near the upper kinematic limit, close to the ρ0 pole mass. The isovector nature of the produced π+π- pairs is also supported by the nonobservation of χc1(3872)→π0π0J/ψ decays. Since no charged partners of the χc1(3872) state have been observed in the ρ±J/ψ decay mode [2,3], the χc1(3872) particle is predominantly an isosinglet state; the J/ψ isospin is also zero. Together, this makes the χc1(3872)→ρ0J/ψ decay isospin violating. The χc1(3872) spin and parities, JPC=1++ [4,5], match the 23P1 excitation of the cc¯ system predicted in the relevant mass range [6]. However, isospin violating decays of charmonium states are highly suppressed. Therefore, quantifying the isospin violation in χc1(3872)→ρ0J/ψ decays is important to understand the nature of the χc1(3872) state, which is under intense debate [7,8]. Isospin conserving χc1(3872)→ωJ/ψ decays provide a suitable normalization process. Such decays have been recently established with a significance of 5σ [9] using the dominant ω→π+π-π0 decay, which has a branching fraction (B) of (89.2±0.7)% [10]. Averaged with earlier measurements [11,12], B(χc1(3872)→ωJ/ψ)/B(χc1(3872)→π+π-J/ψ)=1.4±0.3. However, ω→π+π- decays, with B(ω→π+π-)=(1.53±0.12)% [10], are expected to contribute to the denominator at the 2% level if ρ0-ω interference is neglected. The interference can change this estimate by a large factor. Therefore, an analysis of the mπ+π- spectrum is necessary to disentangle the ρ0 and ω contributions. Such analyses were performed by the CDF [13] and the Belle [3] Collaborations, using the Breit–Wigner sum model, yielding inconclusive results due to the large statistical uncertainties.

In this paper, we report an analysis of a B+→K+χc1(3872), χc1(3872)→π+π-J/ψ, J/ψ→μ+μ- data sample collected using the LHCb detector, with proton-proton (pp) collision energies of 7, 8 and 13 TeV, corresponding to a total integrated luminosity of 9  fb-1. The inclusion of charge-conjugate processes is implied throughout. This sample is about 6 times more sensitive to an ω contribution than those used in Refs. [3,13]. The LHCb detector [14,15] is a single-arm forward spectrometer covering the pseudorapidity range 2<η<5, designed for the study of particles containing b or c quarks. The detector elements that are particularly relevant to this analysis are: a silicon-strip vertex detector surrounding the pp interaction region that allows b hadrons to be identified from their characteristically long flight distance; a tracking system that provides a measurement of the momentum, p, of charged particles; two ring-imaging Cherenkov detectors that are able to discriminate between different species of charged hadrons and the muon detector.

The selection of χc1(3872)→π+π-J/ψ candidates is based on the reconstruction of B+→K+π+π-J/ψ(→μ+μ-) decays, which provides for efficient background suppression and optimal dipion mass resolution. Both muon candidates are identified by the muon detector. The dimuon mass must be consistent with the known J/ψ mass [10], and all five final state particles must form a good-quality vertex significantly displaced from the closest primary pp interaction vertex (PV). The hadron candidate most likely to be a kaon is selected as the kaon candidate. Each hadron must have a significant impact parameter with respect to any PV. To remove multiple entries per event, the B+ candidate with the largest scalar sum of the hadron and J/ψ candidate transverse-momenta is selected. To improve mass resolution, the B+ candidates are kinematically constrained to point to the closest PV and reproduce the known J/ψ mass [16]. The mass of these candidates must be consistent with the known B+ mass [10], which is then also included among the kinematic constraints. The resulting mπ+π-J/ψ distribution (Fig. 1) is fit with a signal (background) shape modeled by a double-sided Crystal Ball [17] (quadratic) function, yielding 6788±117 χc1(3872)→π+π-J/ψ decays with a χc1(3872) mass resolution of σm=2.66±0.09  MeV (natural units are used throughout). The signal purity is 77% within ±2σm around the χc1(3872) mass. The dominant source of background is from B+ decays to J/ψ meson and excited kaons (K*+), which decay to K+π+π-. The dipion mass distribution is obtained by two-dimensional unbinned fits of the χc1(3872) signal yields to the (mπ+π-J/ψ,mπ+π-) data in mπ+π- intervals. The signal shape in mπ+π-J/ψ is fixed to the global fit result of Fig. 1, while the background shape may vary in each mπ+π- interval. The signal and the background mπ+π-J/ψ shapes are then multiplied by a two-body phase-space factor, i.e. the J/ψ momentum in the χc1(3872) rest frame (pJ/ψ), which depends on both mπ+π-J/ψ and mπ+π-. The resultant shapes are normalized to unity using their integral over the fitted phase-space range. The obtained data points are shown in Fig. 2.

110.1103/PhysRevD.108.L011103.f1

Distribution of mπ+π-J/ψ [mK+π+π-J/ψ in the inset] from B+→K+π+π-J/ψ candidates within ±2σ around the B+ [χc1(3872)] mass, overlaid with the fit including the total (solid blue), signal (dashed red), and background (dashed green) components.

210.1103/PhysRevD.108.L011103.f2

Distribution of mπ+π- in χc1(3872)→π+π-J/ψ decays, fit with the ρ0-only model.

Signal simulation is used to obtain the mπ+π- dependence of the dipion mass resolution and of the reconstruction efficiency. In the simulation, pp collisions are generated using pythia [18,19] with a specific LHCb configuration [20]. Decays of unstable particles are described by evtgen [21], in which final-state radiation is generated using photos [22]. In particular, B+→K+χc1(3872), χc1(3872)→J/ψρ0, ρ0→π+π-, J/ψ→μ+μ- signal decays are simulated using the helicity model where the χc1(3872) decays via an S-wave, which describes the angular distributions in the data well [5]. When integrating over mπ+π-, the generated mπ+π- distribution is weighted to reproduce the observed distribution. The interaction of the generated particles with the detector, and its response, are implemented using the geant4 toolkit [23,24] as described in Ref. [25]. The underlying pp interaction is reused multiple times, with an independently generated signal decay for each event [26]. The transverse-momentum distribution of the signal B+ is weighted to match the data. The relative signal reconstruction efficiency obtained from the simulation is well described by a quadratic function, ε(mπ+π-)=0.966+1.345×10-3(mπ+π--700)+1.607×10-6(mπ+π--700)2, and the dipion mass resolution by σ(mπ+π-)=2.39(1-exp(-mπ+π-/220.3))-5.4exp(-mπ+π-/220.3), where mπ+π- and σ(mπ+π-) are in MeV. The simulation underestimates the χc1(3872) and B+ mass resolution by 6% and 14%, respectively. Therefore, σ(mπ+π-) is scaled up by fm=1.06, which is varied from 1.00 to 1.14 to assess a systematic uncertainty. All theoretical probability density functions fit to the data, P(mπ+π-), are multiplied by the relative efficiency function, ε(mπ+π-) and convolved with a Gaussian distribution characterized by a root-mean-square value of fmσ(mπ+π-).

The matrix element, M, describing the three-body decay, χc1(3872)→π+π-J/ψ, is related to P(mπ+π-) via a scaling (S) and the phase-space factors, P(mπ+π-)=SpJ/ψpπ|M|2, where pπ is the pion momentum in the rest frame of the π+π- system. The scaling factor S is a nuisance parameter.

Following the previous analyses [3,13], the ρ0 component is first parametrized as a Breit–Wigner (BW) amplitude, M=BW(s|mρ,Γρ)=mρΓρB1(pπ)/B1(pπρ)mρ2-s-imρΓ(s),Γ(s)=ΓρpπpπρmρsB1(pπ)B1(pπρ),where s≡mπ+π-2, pπρ=pπ(mρ), mρ=775.26  MeV, Γρ=147.4  MeV [10], and B1(p)=p2/[(1+(Rp)2)] is the Blatt–Weisskopf barrier factor for the P-wave decay of a vector particle to π+π- and contains an effective hadron-size parameter R. With R adjusted to 1.45  GeV-1, a complex phase of the BW amplitude varies in the fitted mπ+π- range within 1 degree of the isovector π+π-  P-wave parametrization extracted from the scattering data by the phenomenological analysis of Ref. [27]. A similar value of 1.5  GeV-1 was used in the previous analyses [3,13]. The χ2 fit, with S as the only fit parameter, fails to describe the data as shown in Fig. 2, with a χ2 value per number of degrees of freedom (χ2/n.d.o.f.) equal to 366.6/34. A large disagreement between the data and the χc1(3872)→ρ0J/ψ amplitude at high dipion masses was missed in the previous comparisons with the χc1(3872)→ρ0J/ψ simulation (see Fig. S2 in the Supplemental Material of Ref. [5]; see also Refs. [28,29]), because the evtgen [21] model does not correctly simulate the impact of the phase-space factors (here pJ/ψpπ) on resonant shapes. As a consequence, the large ρ0-ω interference in the data (see below) was mistakenly interpreted as a part of the ρ0 resonance itself.

Adding an ω contribution via the BW sum model with M=BW(s|mρ,Γρ)+aexp(iϕ)BW(s|mω,Γω), where a and ϕ are the amplitude and phase of the ω component relative to the ρ0 term [3,13], improves the fit quality substantially, χ2/n.d.o.f.=102.9/33, yet not enough to be acceptable. Summing BW amplitudes results in matrix elements that are not unitary, violating first principles of scattering theory. A two-channel K-matrix (K) model [30], coupling the π+π- and π+π-π0 channels via an ω contribution, resolves this issue, K=1mρ2-s(gρ→2π2000)+1mω2-s(gω→2π2gω→2πgω→3πgω→2πgω→3πgω→3π2),where g are the coupling constants described later. The gρ→3π2 coupling is 4–5 orders of magnitude smaller than gρ→2π2 [10,31,32] and has been neglected here. The T-matrix is obtained from T^=[1-iKϱ]-1K, where the phase-space matrix ϱ is diagonal, ϱ=diag(ϱ2π(s),ϱ3π(s)) and is detailed in the Supplemental Material [33]. The decay amplitude is given by M=A^2πB1(pπ),where A^2π≡α2πT^2π,2π+α3πT^2π,3π, and the elements of the production Q-vector (α2π, α3π) are real [34]. The coupling constants are fully determined from other experiments [10], gρ→2π2=mρΓρ/ϱ2π(mρ2),gω→3π2=mωΓωB(ω→π+π-π0)/ϱ3π(mω2),gω→2π2=mωΓωB(ω→π+π-)/ϱ2π(mω2).Numerically, gω→2π2/gρ→2π2≈0.0009, while gω→2πgω→3π/gρ→2π2≈0.01. Thus, the diagonal ω coupling to two pions can be neglected in comparison with the off diagonal coupling. In this approximation, equivalent to the full formalism given the precision of this analysis, T^2π,2π≈gρ→2π2mρ2-s-igρ→2π2ϱ2π(s),which is the ρ0 Breit-Wigner amplitude [Eq. (1)] and α2π is the ρ0 production factor in the χc1(3872) decay. A mild dependence of α2π on s is possible, since α2π(s) must be analytic and cannot change much within the resonance widths (see e.g. Ref. [35]). Using Chebyshev polynomials (Cn) to express the dependence: α2π(s)=∑n=0n=NPnCn(s^), where s^≡2(s-smin)/(smax-smin)-1, smin=(380  MeV)2, and smax=(775  MeV)2. The polynomial coefficients Pn are determined from a fit to the data, except for P0, which is set to unity as the normalization choice. For α2π(s) to have the expected theoretical behavior, the series must converge, |Pn+1|<|Pn|, and N should be kept small.

The ω contribution enters via the element, T^2π,3π≈gω→2πgω→3π(mρ2-s)(mρ2-s-igρ→2π2ϱ2π(s))(mω2-s-igω→3π2ϱ3π(s)).This term approaches zero at the bare ρ0 pole mass, which complicates probing the ω contribution, mω=782.66±0.13  MeV and Γω=8.68±0.13  MeV [10]. This zero is an artifact of the K-matrix approach, rather than an expectation from scattering theory. To remove it and restore a more physical behavior of the ω term, α3π=aωmω2-mρ2mρ2-s,is set. The constant term mω2-mρ2 is introduced above to make aω express the ratio of ω/ρ amplitudes at the ω pole. The value of aω is determined by the fit to the data. Using Eqs. (7)–(9), with α2π constant (N=0), is the common method to describe ρ0-ω interference in analyses of the π+π- system [36]. The exact K-matrix formulas are used here. In view of Eq. (7), models with α3π=0 are interpreted as containing a ρ0 component only. The following integrals are calculated to quantify a relative rate of the ω contribution: Itot=∫P(mπ+π-)dmπ+π-, Iρ=∫P(mπ+π-|α3π=0)dmπ+π-, and Iω=∫P(mπ+π-|α2π=0)dmπ+π-, where P(mπ+π-) is neither convolved with the mass resolution, nor multiplied by the efficiency function. The integration is performed over the full phase space. To quantify the overall impact of ω on the total rate, including ρ0-ω interference effects, the ratio Rωall≡1-Iρ/Itot is defined. The traditional ω fit fraction is given by Rω0≡Iω/Itot. Finally, to probe the ratio of the χc1(3872) isospin conserving to isospin violating couplings, the ratio Rω/ρ0≡Iω/Iρ is introduced. Using the B+→K+χc1(3872) simulation, the extraction of the dipion mass distribution and subsequent fit with the K-matrix model, with the efficiency and the mass resolution corrections included, are verified to a precision an order of magnitude better than the total systematic uncertainties reported below.

Allowing the ω term in the K-matrix model with the constant α2π improves the fit quality to χ2/n.d.o.f.=55.1/33, resulting in a significance for the ω contribution of nσ=(χaω=02-χaω≠02)1/2=17.7σ [37]. However, the p-value of the fit is marginal (0.9%). Introducing a term linear in s to α2π(s) improves the fit with χ2/n.d.o.f.=24.7/32, p-value =82%, and a small slope coefficient, P1=0.23±0.05. This fit, shown in Fig. 3, is taken as the default model and yields nσ=8.1σ, Rωall=0.214±0.023, Rω0=0.0193±0.0044, Rω/ρ0=0.0246±0.0062, and aω=0.208±0.024. The ω contribution is at the level of Rω0≈2% as expected from the χc1(3872)→ωJ/ψ, ω→π+π-π0 measurements (see above). The impact on the overall χc1(3872)→π+π-J/ψ rate given by Rωall is an order of magnitude larger, enhanced by the ρ0-ω interference. Allowing a quadratic term in α2π(s) lowers the p-value of the fit to 78% with χ2/n.d.o.f.=24.6/31, indicating that too much freedom is added to the model. The P2 coefficient is small and consistent with zero, P2=0.016±0.047, while all other fit results remain consistent with the default fit, P1=0.21±0.07, Rωall=0.206±0.035, Rω0=0.0178±0.0062, Rω/ρ0=0.0225±0.088, and aω=0.197±0.042. The significance of the ω contribution remains very high, nσ=5.5σ and is now underestimated, since the null hypothesis (aω=0) gives an unphysical polynomial correction (the quadratic term, P2=0.17±0.02, comparable to, and more significant than the linear term, P1=0.16±0.05).

310.1103/PhysRevD.108.L011103.f3

Distribution of mπ+π- in χc1(3872)→π+π-J/ψ decays, fit with the default K-matrix model.

A number of analysis variations have been performed to evaluate systematic uncertainties on the ω fraction, as summarized in Table I. In the dipion mass extraction from the data, Gaussian functions are used for the χc1(3872) shape in mπ+π-J/ψ data, rather than Crystal Ball functions. A cubic polynomial replaces the quadratic function in the relative efficiency parametrization.

I10.1103/PhysRevD.108.L011103.t1

Results for ω fractions with systematic uncertainties. See text for a description of the individual entries.

Fit typeχ2/n.d.o.f.p-valueRωallRω0Rω/ρ0nσDefault24.7/320.820.214±0.0230.019±0.0040.025±0.0068.1σP2≠024.6/310.780.206±0.0350.018±0.0060.023±0.0095.5σGaussian χc1(3872)20.0/320.950.194±0.0240.016±0.0040.020±0.0067.3σcubic ε(mπ+π-)24.5/320.830.221±0.0230.021±0.0050.027±0.0078.1σhad.ID corrections24.6/320.820.214±0.0230.019±0.0040.025±0.0068.1σBDT selection24.6/320.820.207±0.0220.018±0.0040.023±0.0067.9σσ(mπ+π-)×1.026.6/320.740.213±0.0230.019±0.0040.025±0.0068.1σσ(mπ+π-)×1.1422.6/320.890.215±0.0230.020±0.0040.026±0.0068.1σmπ+π-<775  MeV18.0/310.970.196±0.0240.016±0.0040.021±0.0067.1σcosθX<026.9/320.720.211±0.0350.019±0.0070.024±0.0105.2σcosθX>042.2/320.110.217±0.0300.021±0.0060.027±0.0094.2σNR production of 2π24.7/320.820.214±0.0220.019±0.0040.025±0.0068.1σD-wave free24.5/310.790.210±0.0290.017±0.0050.021±0.0077.8σD-wave fixed at 4%24.5/320.820.208±0.0230.018±0.0040.023±0.0067.9σρ′25.1/320.800.209±0.0230.018±0.0040.024±0.0068.1σRprod=0  GeV-124.7/320.820.209±0.0230.019±0.0040.024±0.0067.9σRprod=30  GeV-124.6/320.820.229±0.0220.021±0.0040.028±0.0068.7σR=1.3  GeV-124.7/320.820.216±0.0220.020±0.0040.026±0.0068.2σR=1.6  GeV-124.7/320.820.212±0.0230.019±0.0040.025±0.0068.0σGS model24.8/320.810.221±0.0240.021±0.0050.028±0.0077.8σ Summary0.214±0.023±0.0200.019±0.004±0.0030.025±0.006±0.005>7.1σ

Data-driven corrections to the simulation of the hadron identification are implemented, as a cross-check. To check further the relative efficiency simulation and the background subtraction, a tighter data selection is implemented with a Boosted Decision Tree (BDT) classifier [38], using inputs from hadron identification variables, the B+ vertex quality, the PV impact parameters of the final-state particles and the B+ candidate, the hadron transverse-momenta and the B+ flight distance. The tighter selection reduces the combinatoric background under the B+ peak in the mK+π+π-J/ψ distribution from 9.4% to 3.1%, resulting in an overall background reduction under the χc1(3872) peak in the mπ+π-J/ψ distribution from 23% to 17%. At the same time, the χc1(3872) signal yield is reduced by only 0.4%.

To evaluate uncertainties due to the mass resolution used in the fit, the scaling factor fm is varied and the fit range is reduced to exclude the last interval of the mπ+π- distribution (775–780 MeV), since this falls beyond the phase-space limit, and therefore its content is very sensitive to σ(mπ+π-).

To check for interference effects between B+→K+χc1(3872) and B+→K*+J/ψ decays, the data are split into two subsamples depending on the sign of cosθX, where θX is the χc1(3872) helicity angle, the angle between the K+ and J/ψ momenta in the χc1(3872) rest frame. The composition of K*+ resonances is different in each subsample.

As a variation of the production model, nonresonant (NR) terms are added to the production vector, (A^2π,A^3π)=[1−iKϱ]−1[K(α2π,α3π)+(f2π,f3π)]. Without χc1(3872)→3πJ/ψ data in the fit, the NR production parameter f3π cannot be probed, thus it is set to zero. A good-quality fit is obtained with constant α2π, which gives f2π=(-9.7±1.6)×10-7.

The default fit assumes an S-wave χc1(3872) decay. When a D-wave component is allowed in the fit, multiplying |M|2 by 1+AD2B2(pJ/ψ)/B2(pJ/ψρ), where pJ/ψρ≡pJ/ψ(mρ) and B2(p)≡p4/(9+3(Rp)2+(Rp)4), the AD parameter is consistent with zero, 0.13±0.41. Tuning AD to 0.176 produces a 4% D-wave fraction, equal to the upper limit from studies of the angular correlations [5].

Since the hadron size parameter R is tuned to the scattering data, which automatically includes any ρ0 excitations, there is no strong motivation to include the ρ(1450) pole [10] (ρ′) in the K-matrix model. When included, the fit quality is slightly reduced.

The size parameter R used in B1(pπ) of Eq. (3) could be related to the χc1(3872) size, rather than the ρ0 or ω sizes. Thus, fit variations are tried in which it has an independent value, Rprod, varied from zero to 30  GeV-1. The R value itself is varied within the range 1.3–1.6  GeV-1. An alternative model of the ρ0 shape, that does not include an R-dependent Blatt-Weisskopf form factor, is provided by the Gounaris-Sakurai (GS) formula [39], and was utilized by the BABAR Collaboration to describe high statistics e+e-→π+π-(γ) data [40]. As discussed in the Supplemental Material [33], when applying this prescription with ρ0, ρ′ and ω contributions, an excellent fit to the data is obtained, χ2/n.d.o.f.=24.8/32, p-value =81%, matching the fit quality of the default fit.

Total systematic uncertainties, given at the bottom of Table I, are set to cover the maximal deviation from the default fit results, and are comparable to the statistical uncertainties. The lowest ω significance is 7.1σ, excluding the subsamples and P2≠0 fit as discussed above. This is a more significant observation of χc1(3872)→ωJ/ψ decays than achieved using the dominant ω→π+π-π0 decay channel.

In addition to the resonant coupling constants, the limited phase space in χc1(3872)→π+π-J/ψ decays has a large impact on the Rω/ρ0 value capturing a much smaller fraction of the ω resonance than for the ρ0 resonance. To probe the resonant coupling constants, the phase space can be artificially extended to contain both resonance peaks by setting mχc1(3872)=4000  MeV, as illustrated in the Supplemental Material [33]. Integrating the default P(mπ+π-) model in the extended phase space, Rω/ρ0 ′=0.18±0.05 is obtained and then used to deduce the ratio of the isospin violating to isospin conserving χc1(3872) couplings, gχc1(3872)→ρ0J/ψgχc1(3872)→ωJ/ψ=B(ω→π+π-)Rω/ρ0′=0.29±0.04.This value is an order of magnitude larger than expected for pure cc¯ states, as exemplified by the corresponding value for the ψ(2S) state [10], gψ(2S)→π0J/ψgψ(2S)→ηJ/ψ=B(ψ(2S)→π0J/ψ)B(ψ(2S)→ηJ/ψ)pη3pπ03=0.045±0.001,where pη and pπ0 are the breakup momenta [41]. Therefore, the χc1(3872) state cannot be a pure charmonium state. Large isospin violation is naturally expected in models in which the χc1(3872) state has a significant DD¯* component, either as constituents (i.e. in the “molecular model”) or generated dynamically in the decay [42–51]. The proximity of the χc1(3872) mass to the D0D¯*0 threshold, enhances such contributions over D+D*- combinations, which are 8 MeV heavier. It has also been suggested in compact tetraquark models that two neutral states could be degenerate and mix, giving rise to large isospin violation in χc1(3872) decays [52–54].

In summary, the ρ0 and ω contributions to χc1(3872)→π+π-J/ψ decays are resolved for the first time using a much larger data sample than previously available. Through ρ0-ω interference, the ω contribution accounts for (21.4±2.3±2.0)% of the total rate, or equivalently, B(χc1(3872)→ρ0J/ψ)/B(χc1(3872)→π+π-J/ψ)=(78.6±2.3±2.0)%. Excluding interference effects, the ω contribution, (1.9±0.4±0.3)%, is found to be consistent with, but more precise than the previous χc1(3872)→ωJ/ψ measurements utilizing ω→π+π-π0 decays [9,11,12]. The isospin violating ρ0 contribution, quantified for the first time with proper subtraction of the ω contribution, is an order of magnitude too large for χc1(3872) to be a pure charmonium state.

This analysis also serves as an excellent illustration for why a simple sum of ρ0 and ω Breit-Wigner amplitudes should not be used to describe interfering resonances. Such a model fails to describe the data, while the K-matrix approach, which respects unitarity, provides an excellent description of the data and gives numerical results that are consistent with the previous χc1(3872)→ωJ/ψ measurements utilizing ω→π+π-π0 decays.

We express our gratitude to our colleagues in the CERN accelerator departments for the excellent performance of the LHC. We thank the technical and administrative staff at the LHCb institutes. We acknowledge support from CERN and from the national agencies: CAPES, CNPq, FAPERJ, and FINEP (Brazil); MOST and NSFC (China); CNRS/IN2P3 (France); BMBF, DFG and MPG (Germany); INFN (Italy); NWO (Netherlands); MNiSW and NCN (Poland); MEN/IFA (Romania); MICINN (Spain); SNSF and SER (Switzerland); NASU (Ukraine); STFC (United Kingdom); DOE NP and NSF (USA). We acknowledge the computing resources that are provided by CERN, IN2P3 (France), KIT and DESY (Germany), INFN (Italy), SURF (Netherlands), PIC (Spain), GridPP (United Kingdom), CSCS (Switzerland), IFIN-HH (Romania), CBPF (Brazil), Polish WLCG (Poland) and NERSC (USA). We are indebted to the communities behind the multiple open-source software packages on which we depend. Individual groups or members have received support from ARC and ARDC (Australia); Minciencias (Colombia); AvH Foundation (Germany); EPLANET, Marie Skłodowska-Curie Actions and ERC (European Union); A*MIDEX, ANR, IPhU and Labex P2IO, and Région Auvergne-Rhône-Alpes (France); Key Research Program of Frontier Sciences of CAS, CAS PIFI, CAS CCEPP, Fundamental Research Funds for the Central Universities, and Sci. and Tech. Program of Guangzhou (China); MEFP, GVA, XuntaGal, GENCAT and Prog. Atracción Talento, CM (Spain); SRC (Sweden); the Leverhulme Trust, the Royal Society and UKRI (United Kingdom).

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