EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)

 CERN-PH-EP-2013-004 LHCb-PAPER-2012-045 January 24, 2013

Analysis of the resonant components in $\kern
3.73305pt\overline{\kern-3.73305ptB}{}^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\pi^{+}\pi^{-}$

The LHCb collaboration†††Authors are listed on the following pages.

Interpretation of $C\\!P$ violation measurements using charmonium decays, in
both the $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ and $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ systems, can be subject to
changes due to “penguin” type diagrams. These effects can be investigated
using measurements of the Cabibbo-suppressed $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\pi^{+}\pi^{-}$ decays. The final state composition of this channel is
investigated using a 1.0 fb-1 sample of data produced in 7 TeV $pp$ collisions
at the LHC and collected by the LHCb experiment. A modified Dalitz plot
analysis is performed using both the invariant mass spectra and the decay
angular distributions. An improved measurement of the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\pi^{+}\pi^{-}$ branching fraction of $(3.97\pm 0.09\pm 0.11\pm
0.16)\times 10^{-5}$ is reported where the first uncertainty is statistical,
the second is systematic and the third is due to the uncertainty of the
branching fraction of the decay
$B^{-}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{-}$ used as a
normalization channel. Significant production of $f_{0}(500)$ and $\rho(770)$
resonances is found in the substructure of the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\pi^{+}\pi^{-}$ final state, and
this indicates that they are viable final states for $C\\!P$ violation
studies. In contrast evidence for the $f_{0}(980)$ resonance is not found.
This allows us to establish the first upper limit on the branching fraction
product ${\cal{B}}\left(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}f_{0}(980)\right)\times{\cal{B}}\left(f_{0}(980)\rightarrow\pi^{+}\pi^{-}\right)<1.1\times
10^{-6}$, leading to an upper limit on the absolute value of the mixing angle
of the $f_{0}(980)$ with the $f_{0}(500)$ of less than $31^{\circ}$, both at
90% confidence level.

Submitted to Physical Review D

© CERN on behalf of the LHCb collaboration, license CC-BY-3.0.

LHCb collaboration

R. Aaij38, C. Abellan Beteta33,n, A. Adametz11, B. Adeva34, M. Adinolfi43, C.
Adrover6, A. Affolder49, Z. Ajaltouni5, J. Albrecht9, F. Alessio35, M.
Alexander48, S. Ali38, G. Alkhazov27, P. Alvarez Cartelle34, A.A. Alves
Jr22,35, S. Amato2, Y. Amhis7, L. Anderlini17,f, J. Anderson37, R.
Andreassen57, R.B. Appleby51, O. Aquines Gutierrez10, F. Archilli18, A.
Artamonov 32, M. Artuso53, E. Aslanides6, G. Auriemma22,m, S. Bachmann11, J.J.
Back45, C. Baesso54, V. Balagura28, W. Baldini16, R.J. Barlow51, C.
Barschel35, S. Barsuk7, W. Barter44, Th. Bauer38, A. Bay36, J. Beddow48, I.
Bediaga1, S. Belogurov28, K. Belous32, I. Belyaev28, E. Ben-Haim8, M.
Benayoun8, G. Bencivenni18, S. Benson47, J. Benton43, A. Berezhnoy29, R.
Bernet37, M.-O. Bettler44, M. van Beuzekom38, A. Bien11, S. Bifani12, T.
Bird51, A. Bizzeti17,h, P.M. Bjørnstad51, T. Blake35, F. Blanc36, C. Blanks50,
J. Blouw11, S. Blusk53, A. Bobrov31, V. Bocci22, A. Bondar31, N. Bondar27, W.
Bonivento15, S. Borghi51, A. Borgia53, T.J.V. Bowcock49, E. Bowen37, C.
Bozzi16, T. Brambach9, J. van den Brand39, J. Bressieux36, D. Brett51, M.
Britsch10, T. Britton53, N.H. Brook43, H. Brown49, I. Burducea26, A.
Bursche37, J. Buytaert35, S. Cadeddu15, O. Callot7, M. Calvi20,j, M. Calvo
Gomez33,n, A. Camboni33, P. Campana18,35, A. Carbone14,c, G. Carboni21,k, R.
Cardinale19,i, A. Cardini15, H. Carranza-Mejia47, L. Carson50, K. Carvalho
Akiba2, G. Casse49, M. Cattaneo35, Ch. Cauet9, M. Charles52, Ph.
Charpentier35, P. Chen3,36, N. Chiapolini37, M. Chrzaszcz 23, K. Ciba35, X.
Cid Vidal34, G. Ciezarek50, P.E.L. Clarke47, M. Clemencic35, H.V. Cliff44, J.
Closier35, C. Coca26, V. Coco38, J. Cogan6, E. Cogneras5, P. Collins35, A.
Comerma-Montells33, A. Contu15, A. Cook43, M. Coombes43, G. Corti35, B.
Couturier35, G.A. Cowan36, D. Craik45, S. Cunliffe50, R. Currie47, C.
D’Ambrosio35, P. David8, P.N.Y. David38, I. De Bonis4, K. De Bruyn38, S. De
Capua51, M. De Cian37, J.M. De Miranda1, L. De Paula2, W. De Silva57, P. De
Simone18, D. Decamp4, M. Deckenhoff9, H. Degaudenzi36,35, L. Del Buono8, C.
Deplano15, D. Derkach14, O. Deschamps5, F. Dettori39, A. Di Canto11, J.
Dickens44, H. Dijkstra35, P. Diniz Batista1, M. Dogaru26, F. Domingo
Bonal33,n, S. Donleavy49, F. Dordei11, A. Dosil Suárez34, D. Dossett45, A.
Dovbnya40, F. Dupertuis36, R. Dzhelyadin32, A. Dziurda23, A. Dzyuba27, S.
Easo46,35, U. Egede50, V. Egorychev28, S. Eidelman31, D. van Eijk38, S.
Eisenhardt47, U. Eitschberger9, R. Ekelhof9, L. Eklund48, I. El Rifai5, Ch.
Elsasser37, D. Elsby42, A. Falabella14,e, C. Färber11, G. Fardell47, C.
Farinelli38, S. Farry12, V. Fave36, D. Ferguson47, V. Fernandez Albor34, F.
Ferreira Rodrigues1, M. Ferro-Luzzi35, S. Filippov30, C. Fitzpatrick35, M.
Fontana10, F. Fontanelli19,i, R. Forty35, O. Francisco2, M. Frank35, C.
Frei35, M. Frosini17,f, S. Furcas20, E. Furfaro21, A. Gallas Torreira34, D.
Galli14,c, M. Gandelman2, P. Gandini52, Y. Gao3, J. Garofoli53, P. Garosi51,
J. Garra Tico44, L. Garrido33, C. Gaspar35, R. Gauld52, E. Gersabeck11, M.
Gersabeck51, T. Gershon45,35, Ph. Ghez4, V. Gibson44, V.V. Gligorov35, C.
Göbel54, D. Golubkov28, A. Golutvin50,28,35, A. Gomes2, H. Gordon52, M.
Grabalosa Gándara5, R. Graciani Diaz33, L.A. Granado Cardoso35, E. Graugés33,
G. Graziani17, A. Grecu26, E. Greening52, S. Gregson44, O. Grünberg55, B.
Gui53, E. Gushchin30, Yu. Guz32, T. Gys35, C. Hadjivasiliou53, G. Haefeli36,
C. Haen35, S.C. Haines44, S. Hall50, T. Hampson43, S. Hansmann-Menzemer11, N.
Harnew52, S.T. Harnew43, J. Harrison51, P.F. Harrison45, T. Hartmann55, J.
He7, V. Heijne38, K. Hennessy49, P. Henrard5, J.A. Hernando Morata34, E. van
Herwijnen35, E. Hicks49, D. Hill52, M. Hoballah5, C. Hombach51, P. Hopchev4,
W. Hulsbergen38, P. Hunt52, T. Huse49, N. Hussain52, D. Hutchcroft49, D.
Hynds48, V. Iakovenko41, P. Ilten12, R. Jacobsson35, A. Jaeger11, E. Jans38,
F. Jansen38, P. Jaton36, F. Jing3, M. John52, D. Johnson52, C.R. Jones44, B.
Jost35, M. Kaballo9, S. Kandybei40, M. Karacson35, T.M. Karbach35, I.R.
Kenyon42, U. Kerzel35, T. Ketel39, A. Keune36, B. Khanji20, O. Kochebina7, I.
Komarov36,29, R.F. Koopman39, P. Koppenburg38, M. Korolev29, A. Kozlinskiy38,
L. Kravchuk30, K. Kreplin11, M. Kreps45, G. Krocker11, P. Krokovny31, F.
Kruse9, M. Kucharczyk20,23,j, V. Kudryavtsev31, T. Kvaratskheliya28,35, V.N.
La Thi36, D. Lacarrere35, G. Lafferty51, A. Lai15, D. Lambert47, R.W.
Lambert39, E. Lanciotti35, G. Lanfranchi18,35, C. Langenbruch35, T. Latham45,
C. Lazzeroni42, R. Le Gac6, J. van Leerdam38, J.-P. Lees4, R. Lefèvre5, A.
Leflat29,35, J. Lefrançois7, O. Leroy6, Y. Li3, L. Li Gioi5, M. Liles49, R.
Lindner35, C. Linn11, B. Liu3, G. Liu35, J. von Loeben20, J.H. Lopes2, E.
Lopez Asamar33, N. Lopez-March36, H. Lu3, J. Luisier36, H. Luo47, F.
Machefert7, I.V. Machikhiliyan4,28, F. Maciuc26, O. Maev27,35, S. Malde52, G.
Manca15,d, G. Mancinelli6, N. Mangiafave44, U. Marconi14, R. Märki36, J.
Marks11, G. Martellotti22, A. Martens8, L. Martin52, A. Martín Sánchez7, M.
Martinelli38, D. Martinez Santos39, D. Martins Tostes2, A. Massafferri1, R.
Matev35, Z. Mathe35, C. Matteuzzi20, M. Matveev27, E. Maurice6, A.
Mazurov16,30,35,e, J. McCarthy42, R. McNulty12, B. Meadows57,52, F. Meier9, M.
Meissner11, M. Merk38, D.A. Milanes8, M.-N. Minard4, J. Molina Rodriguez54, S.
Monteil5, D. Moran51, P. Morawski23, R. Mountain53, I. Mous38, F. Muheim47, K.
Müller37, R. Muresan26, B. Muryn24, B. Muster36, P. Naik43, T. Nakada36, R.
Nandakumar46, I. Nasteva1, M. Needham47, N. Neufeld35, A.D. Nguyen36, T.D.
Nguyen36, C. Nguyen-Mau36,o, M. Nicol7, V. Niess5, R. Niet9, N. Nikitin29, T.
Nikodem11, S. Nisar56, A. Nomerotski52, A. Novoselov32, A. Oblakowska-Mucha24,
V. Obraztsov32, S. Oggero38, S. Ogilvy48, O. Okhrimenko41, R. Oldeman15,d,35,
M. Orlandea26, J.M. Otalora Goicochea2, P. Owen50, B.K. Pal53, A. Palano13,b,
M. Palutan18, J. Panman35, A. Papanestis46, M. Pappagallo48, C. Parkes51, C.J.
Parkinson50, G. Passaleva17, G.D. Patel49, M. Patel50, G.N. Patrick46, C.
Patrignani19,i, C. Pavel-Nicorescu26, A. Pazos Alvarez34, A. Pellegrino38, G.
Penso22,l, M. Pepe Altarelli35, S. Perazzini14,c, D.L. Perego20,j, E. Perez
Trigo34, A. Pérez-Calero Yzquierdo33, P. Perret5, M. Perrin-Terrin6, G.
Pessina20, K. Petridis50, A. Petrolini19,i, A. Phan53, E. Picatoste Olloqui33,
B. Pietrzyk4, T. Pilař45, D. Pinci22, S. Playfer47, M. Plo Casasus34, F.
Polci8, G. Polok23, A. Poluektov45,31, E. Polycarpo2, D. Popov10, B.
Popovici26, C. Potterat33, A. Powell52, J. Prisciandaro36, V. Pugatch41, A.
Puig Navarro36, W. Qian4, J.H. Rademacker43, B. Rakotomiaramanana36, M.S.
Rangel2, I. Raniuk40, N. Rauschmayr35, G. Raven39, S. Redford52, M.M. Reid45,
A.C. dos Reis1, S. Ricciardi46, A. Richards50, K. Rinnert49, V. Rives
Molina33, D.A. Roa Romero5, P. Robbe7, E. Rodrigues51, P. Rodriguez Perez34,
G.J. Rogers44, S. Roiser35, V. Romanovsky32, A. Romero Vidal34, J. Rouvinet36,
T. Ruf35, H. Ruiz33, G. Sabatino22,k, J.J. Saborido Silva34, N. Sagidova27, P.
Sail48, B. Saitta15,d, C. Salzmann37, B. Sanmartin Sedes34, M. Sannino19,i, R.
Santacesaria22, C. Santamarina Rios34, E. Santovetti21,k, M. Sapunov6, A.
Sarti18,l, C. Satriano22,m, A. Satta21, M. Savrie16,e, D. Savrina28,29, P.
Schaack50, M. Schiller39, H. Schindler35, S. Schleich9, M. Schlupp9, M.
Schmelling10, B. Schmidt35, O. Schneider36, A. Schopper35, M.-H. Schune7, R.
Schwemmer35, B. Sciascia18, A. Sciubba18,l, M. Seco34, A. Semennikov28, K.
Senderowska24, I. Sepp50, N. Serra37, J. Serrano6, P. Seyfert11, M. Shapkin32,
I. Shapoval40,35, P. Shatalov28, Y. Shcheglov27, T. Shears49,35, L.
Shekhtman31, O. Shevchenko40, V. Shevchenko28, A. Shires50, R. Silva
Coutinho45, T. Skwarnicki53, N.A. Smith49, E. Smith52,46, M. Smith51, K.
Sobczak5, M.D. Sokoloff57, F.J.P. Soler48, F. Soomro18,35, D. Souza43, B.
Souza De Paula2, B. Spaan9, A. Sparkes47, P. Spradlin48, F. Stagni35, S.
Stahl11, O. Steinkamp37, S. Stoica26, S. Stone53, B. Storaci37, M.
Straticiuc26, U. Straumann37, V.K. Subbiah35, S. Swientek9, V. Syropoulos39,
M. Szczekowski25, P. Szczypka36,35, T. Szumlak24, S. T’Jampens4, M.
Teklishyn7, E. Teodorescu26, F. Teubert35, C. Thomas52, E. Thomas35, J. van
Tilburg11, V. Tisserand4, M. Tobin37, S. Tolk39, D. Tonelli35, S. Topp-
Joergensen52, N. Torr52, E. Tournefier4,50, S. Tourneur36, M.T. Tran36, M.
Tresch37, A. Tsaregorodtsev6, P. Tsopelas38, N. Tuning38, M. Ubeda Garcia35,
A. Ukleja25, D. Urner51, U. Uwer11, V. Vagnoni14, G. Valenti14, R. Vazquez
Gomez33, P. Vazquez Regueiro34, S. Vecchi16, J.J. Velthuis43, M. Veltri17,g,
G. Veneziano36, M. Vesterinen35, B. Viaud7, D. Vieira2, X. Vilasis-
Cardona33,n, A. Vollhardt37, D. Volyanskyy10, D. Voong43, A. Vorobyev27, V.
Vorobyev31, C. Voß55, H. Voss10, R. Waldi55, R. Wallace12, S. Wandernoth11, J.
Wang53, D.R. Ward44, N.K. Watson42, A.D. Webber51, D. Websdale50, M.
Whitehead45, J. Wicht35, J. Wiechczynski23, D. Wiedner11, L. Wiggers38, G.
Wilkinson52, M.P. Williams45,46, M. Williams50,p, F.F. Wilson46, J. Wishahi9,
M. Witek23, S.A. Wotton44, S. Wright44, S. Wu3, K. Wyllie35, Y. Xie47,35, F.
Xing52, Z. Xing53, Z. Yang3, R. Young47, X. Yuan3, O. Yushchenko32, M.
Zangoli14, M. Zavertyaev10,a, F. Zhang3, L. Zhang53, W.C. Zhang12, Y. Zhang3,
A. Zhelezov11, A. Zhokhov28, L. Zhong3, A. Zvyagin35.

1Centro Brasileiro de Pesquisas Físicas (CBPF), Rio de Janeiro, Brazil
2Universidade Federal do Rio de Janeiro (UFRJ), Rio de Janeiro, Brazil
3Center for High Energy Physics, Tsinghua University, Beijing, China
4LAPP, Université de Savoie, CNRS/IN2P3, Annecy-Le-Vieux, France
5Clermont Université, Université Blaise Pascal, CNRS/IN2P3, LPC, Clermont-
Ferrand, France
6CPPM, Aix-Marseille Université, CNRS/IN2P3, Marseille, France
7LAL, Université Paris-Sud, CNRS/IN2P3, Orsay, France
8LPNHE, Université Pierre et Marie Curie, Université Paris Diderot,
CNRS/IN2P3, Paris, France
9Fakultät Physik, Technische Universität Dortmund, Dortmund, Germany
10Max-Planck-Institut für Kernphysik (MPIK), Heidelberg, Germany
11Physikalisches Institut, Ruprecht-Karls-Universität Heidelberg, Heidelberg,
Germany
12School of Physics, University College Dublin, Dublin, Ireland
13Sezione INFN di Bari, Bari, Italy
14Sezione INFN di Bologna, Bologna, Italy
15Sezione INFN di Cagliari, Cagliari, Italy
16Sezione INFN di Ferrara, Ferrara, Italy
17Sezione INFN di Firenze, Firenze, Italy
18Laboratori Nazionali dell’INFN di Frascati, Frascati, Italy
19Sezione INFN di Genova, Genova, Italy
20Sezione INFN di Milano Bicocca, Milano, Italy
21Sezione INFN di Roma Tor Vergata, Roma, Italy
22Sezione INFN di Roma La Sapienza, Roma, Italy
23Henryk Niewodniczanski Institute of Nuclear Physics Polish Academy of
Sciences, Kraków, Poland
24AGH University of Science and Technology, Kraków, Poland
25National Center for Nuclear Research (NCBJ), Warsaw, Poland
26Horia Hulubei National Institute of Physics and Nuclear Engineering,
Bucharest-Magurele, Romania
27Petersburg Nuclear Physics Institute (PNPI), Gatchina, Russia
28Institute of Theoretical and Experimental Physics (ITEP), Moscow, Russia
29Institute of Nuclear Physics, Moscow State University (SINP MSU), Moscow,
Russia
30Institute for Nuclear Research of the Russian Academy of Sciences (INR RAN),
Moscow, Russia
31Budker Institute of Nuclear Physics (SB RAS) and Novosibirsk State
University, Novosibirsk, Russia
32Institute for High Energy Physics (IHEP), Protvino, Russia
33Universitat de Barcelona, Barcelona, Spain
34Universidad de Santiago de Compostela, Santiago de Compostela, Spain
35European Organization for Nuclear Research (CERN), Geneva, Switzerland
36Ecole Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland
37Physik-Institut, Universität Zürich, Zürich, Switzerland
38Nikhef National Institute for Subatomic Physics, Amsterdam, The Netherlands
39Nikhef National Institute for Subatomic Physics and VU University Amsterdam,
Amsterdam, The Netherlands
40NSC Kharkiv Institute of Physics and Technology (NSC KIPT), Kharkiv, Ukraine
41Institute for Nuclear Research of the National Academy of Sciences (KINR),
Kyiv, Ukraine
42University of Birmingham, Birmingham, United Kingdom
43H.H. Wills Physics Laboratory, University of Bristol, Bristol, United
Kingdom
44Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom
45Department of Physics, University of Warwick, Coventry, United Kingdom
46STFC Rutherford Appleton Laboratory, Didcot, United Kingdom
47School of Physics and Astronomy, University of Edinburgh, Edinburgh, United
Kingdom
48School of Physics and Astronomy, University of Glasgow, Glasgow, United
Kingdom
49Oliver Lodge Laboratory, University of Liverpool, Liverpool, United Kingdom
50Imperial College London, London, United Kingdom
51School of Physics and Astronomy, University of Manchester, Manchester,
United Kingdom
52Department of Physics, University of Oxford, Oxford, United Kingdom
53Syracuse University, Syracuse, NY, United States
54Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), Rio de
Janeiro, Brazil, associated to 2
55Institut für Physik, Universität Rostock, Rostock, Germany, associated to 11
56Institute of Information Technology, COMSATS, Lahore, Pakistan, associated
to 53
57University of Cincinnati, Cincinnati, OH, United States, associated to 53

aP.N. Lebedev Physical Institute, Russian Academy of Science (LPI RAS),
Moscow, Russia
bUniversità di Bari, Bari, Italy
cUniversità di Bologna, Bologna, Italy
dUniversità di Cagliari, Cagliari, Italy
eUniversità di Ferrara, Ferrara, Italy
fUniversità di Firenze, Firenze, Italy
gUniversità di Urbino, Urbino, Italy
hUniversità di Modena e Reggio Emilia, Modena, Italy
iUniversità di Genova, Genova, Italy
jUniversità di Milano Bicocca, Milano, Italy
kUniversità di Roma Tor Vergata, Roma, Italy
lUniversità di Roma La Sapienza, Roma, Italy
mUniversità della Basilicata, Potenza, Italy
nLIFAELS, La Salle, Universitat Ramon Llull, Barcelona, Spain
oHanoi University of Science, Hanoi, Viet Nam
pMassachusetts Institute of Technology, Cambridge, MA, United States

##  1 Introduction

$C\\!P$ violation measurements using neutral $B$ meson decays into
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ mesons are of prime importance
both for determinations of Standard Model (SM) parameters and searching for
physics beyond the SM. In the case of $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ decays, the final state
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{0}_{\rm\scriptscriptstyle S}$
is the most important for measuring $\sin 2\beta$ [1, *Adachi:2012et,
*:2012ke], while in the case of $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ decays, used to measure
$\phi_{s}$, only the final states ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\phi$ [4, 5, 6, *Abazov:2011ry, *:2012fu], and
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\pi^{+}\pi^{-}$ [9] have been
used so far, where the largest component of the latter is
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}f_{0}(980)$[10, *Stone:2008ak].
The decay rate for these ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ modes
is dominated by the color-suppressed tree level diagram, an example of which
is shown for $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ decays in Fig.
1(a), while penguin processes, an example of which is shown in Fig. 1(b), are
expected to be suppressed. Theoretical predictions on the effects of such
“penguin pollution” vary widely for both $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ and $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ decays [12, *Li:2006vq,
*Boos:2004xp, *Ciuchini:2005mg, *Bhattacharya:2012ph, *Fleischer:1999nz,
*Faller:2008gt], so it is incumbent upon experimentalists to limit possible
changes in the value of the $C\\!P$ violating angles measured using other
decay modes.

Figure 1: (a) Tree level and (b) penguin diagram examples for $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ decays into
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{0}_{\rm\scriptscriptstyle S}$.

The decay $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\pi^{+}\pi^{-}$ can occur via a Cabibbo suppressed tree level diagram,
shown in Fig. 2(a), or via several penguin diagrams. An example is shown in
Fig. 2(b), while others are illustrated in Ref. [19]. These decays are
interesting because they can also be used to measure or limit the amount of
penguin pollution. The advantage in using the decay $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\pi^{+}\pi^{-}$ arises because the relative amount of pollution is
larger. In the allowed decays, e.g. $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{0}_{\rm\scriptscriptstyle S}$, the penguin amplitude is multiplied by
a factor of $\lambda^{2}Re^{i\phi}$, where $\lambda$ is the sine of the
Cabibbo angle $(\approx 0.22)$, while in the suppressed decays the factor
becomes $R^{\prime}e^{i\phi^{\prime}}$, where $R$ and $R^{\prime}$, and $\phi$
and $\phi^{\prime}$ are expected to be similar in size [19]. A similar study
uses the decay $B_{s}^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{0}_{\rm\scriptscriptstyle S}$ [20, *Aaij:2012di, *DeBruyn:2010hh].

Figure 2: (a) Tree level and (b) penguin diagram for $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ decays into
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\pi^{+}\pi^{-}$.

$C\\!P$ violation measurements in the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\pi^{+}\pi^{-}$ mode utilizing $B^{0}-\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ mixing determine $\sin 2\beta^{\rm
eff}$ which can be compared to the well measured $\sin 2\beta$. Differences
can be used to estimate the magnitude of penguin effects. Knowledge of the
final state structure is the first step in this program. Such measurements on
$\sin 2\beta^{\rm eff}$ have been attempted in the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ system by using the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\pi^{0}$ final state [23,
*Aubert:2008bs, *Jung:2012pz].

In order to ascertain the viability of such $C\\!P$ violation measurements we
perform a full “Dalitz like” analysis of the final state. Regions in
$\pi^{+}\pi^{-}$ mass that correspond to spin-0 final states would be $C\\!P$
eigenstates. Final states containing vector resonances, such as the
$\rho(770)$ can be analyzed in a similar manner as was done for the decay
$\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\phi$ [4, 5, 6, *Abazov:2011ry, *:2012fu].

It is also of interest to search for the $f_{0}(980)$ contribution and to
obtain information concerning the mixing angle between the $f_{0}(980)$ and
the $f_{0}(500)$111This particle has been identified previously as the
$f_{0}(600)$ or $\sigma$ resonance. partners in the scalar nonet, as the
latter should couple strongly to the $d\bar{d}$ system. Branching fractions
for $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\pi^{+}\pi^{-}$ and ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\rho^{0}$ have previously been measured by the BaBar collaboration [26,
*Aubert:2007xw].

In this paper the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\pi^{+}$ and
$\pi^{+}\pi^{-}$ mass spectra and decay angular distributions are used to
determine the resonant and non-resonant components. This differs from a
classical Dalitz plot analysis [28] because one of the particles in the final
state, the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ meson, has spin-1
and its three decay amplitudes must be considered. We first show that there
are no evident structures in the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\pi^{+}$ invariant mass, and then model the $\pi^{+}\pi^{-}$ invariant
mass with a series of resonant and non-resonant amplitudes. The data are then
fitted with the coherent sum of these amplitudes. We report on the resonant
structure and the $C\\!P$ content of the final state.

##  2 Data sample and selection requirements

The data sample consists of $1.0~{}\rm fb^{-1}$ of integrated luminosity
collected with the LHCb detector [29] using $pp$ collisions at a center-of-
mass energy of 7 TeV. The detector is a single-arm forward spectrometer
covering the pseudorapidity range $2<\eta<5$, designed for the study of
particles containing $b$ or $c$ quarks. Components include a high precision
tracking system consisting of a silicon-strip vertex detector surrounding the
$pp$ interaction region, a large-area silicon-strip detector located upstream
of a dipole magnet with a bending power of about $4{\rm\,Tm}$, and three
stations of silicon-strip detectors and straw drift-tubes placed downstream.
The combined tracking system has a momentum222We work in units where $c=1$.
resolution $\Delta p/p$ that varies from 0.4% at
5$\mathrm{\,Ge\kern-1.00006ptV}$ to 0.6% at
100$\mathrm{\,Ge\kern-1.00006ptV}$, and an impact parameter resolution of
20$\,\upmu\rm m$ for tracks with large transverse momentum ($p_{\rm T}$) with
respect to the proton beam direction. Charged hadrons are identified using two
ring-imaging Cherenkov (RICH) detectors. Photon, electron and hadron
candidates are identified by a calorimeter system consisting of scintillating-
pad and preshower detectors, an electromagnetic calorimeter and a hadronic
calorimeter. Muons are identified by a system composed of alternating layers
of iron and multiwire proportional chambers. The trigger consists of a
hardware stage, based on information from the calorimeter and muon systems,
followed by a software stage that applies a full event reconstruction [30].

Events are triggered by a ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\\!\rightarrow\mu^{+}\mu^{-}$ decay, requiring two identified muons with
opposite charge, $\mbox{$p_{\rm T}$}(\mu^{\pm})$ greater than
500$\mathrm{\,Me\kern-1.00006ptV}$, an invariant mass within
120$\mathrm{\,Me\kern-1.00006ptV}$ of the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ mass [31], and form a vertex
with a fit $\chi^{2}$ less than 16. After applying these requirements, there
is a large ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ signal over a small
background [32]. Only candidates with dimuon invariant mass between
$-$48$\mathrm{\,Me\kern-1.00006ptV}$ and +43$\mathrm{\,Me\kern-1.00006ptV}$
relative to the observed ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ mass
peak are selected, corresponding a window of about $\pm 3\sigma$. The
requirement is asymmetric because of final state electromagnetic radiation.
The two muons subsequently are kinematically constrained to the known
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ mass.

Other requirements are imposed to isolate $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ candidates with high signal yield
and minimum background. This is accomplished by combining the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\rightarrow\mu^{+}\mu^{-}$
candidate with a pair of pion candidates of opposite charge, and then testing
if all four tracks form a common decay vertex. Pion candidates are each
required to have $p_{\rm T}$ greater than 250$\mathrm{\,Me\kern-1.00006ptV}$,
and the scalar sum of the two transverse momenta, $\mbox{$p_{\rm
T}$}(\pi^{+})+\mbox{$p_{\rm T}$}(\pi^{-})$, must be larger than
900$\mathrm{\,Me\kern-1.00006ptV}$. The impact parameter (IP) is the distance
of closest approach of a track to the primary vertex (PV). To test for
inconsistency with production at the PV, the IP $\chi^{2}$ is computed as the
difference between the $\chi^{2}$ of the PV reconstructed with and without the
considered track. Each pion must have an IP $\chi^{2}$ greater than 9. Both
pions must also come from a common vertex with an acceptable $\chi^{2}$ and
form a vertex with the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ with a
$\chi^{2}$ per number of degrees of freedom (ndf) less than 10 (here ndf
equals five). Pion and kaon candidates are positively identified using the
RICH system. Cherenkov photons are matched to tracks, the emission angles of
the photons compared with those expected if the particle is an electron, pion,
kaon or proton, and a likelihood is then computed. The particle identification
makes use of the logarithm of the likelihood ratio comparing two particle
hypotheses (DLL). For pion selection we require DLL$(\pi-K)>-10$.

The four-track $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ candidate
must have a flight distance of more than 1.5 mm, where the average decay
length resolution is 0.17 mm. The angle between the combined momentum vector
of the decay products and the vector formed from the positions of the PV and
the decay vertex (pointing angle) is required to be less than $2.5^{\circ}$.

Events satisfying this preselection are then further filtered using a
multivariate analyzer based on a Boosted Decision Tree (BDT) technique [33].
The BDT uses six variable that are chosen in a manner that does not introduce
an asymmetry between either the two muons or the two pions. They are the
minimum DLL($\mu-\pi$) of the $\mu^{+}$ and $\mu^{-}$, the minimum $p_{\rm T}$
of the $\pi^{+}$ and $\pi^{-}$, the minimum of the IP $\chi^{2}$ of the
$\pi^{+}$ and $\pi^{-}$, the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ vertex $\chi^{2}$, the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ pointing angle, and the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ flight distance. There is
discrimination power between signal and background in all of these variables,
especially the $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ vertex
$\chi^{2}$.

The background sample used to train the BDT consists of the events in the
$\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ mass sideband having
$5566<m({J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\pi^{+}\pi^{-})<5616$$\mathrm{\,Me\kern-1.00006ptV}$. The signal sample
consists of two million $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}(\rightarrow\mu^{+}\mu^{-})\pi^{+}\pi^{-}$ Monte Carlo simulated events
that are generated uniformly in phase space, using Pythia [34] with a special
LHCb parameter tune [35], and the LHCb detector simulation based on Geant4
[36] described in Ref. [37]. Separate samples are used to train and test the
BDT. The distributions of the BDT classifier for signal and background are
shown in Fig. 3. To minimize a possible bias on the signal acceptance due to
the BDT, we choose a relatively loose requirement of the BDT classifier
$>0.05$ which has a 96% signal efficiency and a 92% background rejection rate.

Figure 3: Distributions of the BDT classifier for both training and test
samples of ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\pi^{+}\pi^{-}$ signal
and background events. The signal samples are from simulation and the
background samples are from data. Figure 4: Invariant mass of
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\pi^{+}\pi^{-}$ combinations. The
data are fitted with a double-Gaussian signal and several background
functions. The (red) solid double-Gaussian function centered at
5280$\mathrm{\,Me\kern-1.00006ptV}$ is the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ signal, the (brown) dotted line
shows the combinatorial background, the (green) short-dashed shows the $B^{-}$
background, the (purple) dot-dashed line shows the contribution of $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\pi^{+}\pi^{-}$ decays, the (black) dot-long dashed is the sum of $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\eta^{\prime}(\rightarrow\rho\gamma)$ and $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\phi(\rightarrow\pi^{+}\pi^{-}\pi^{0})$ backgrounds, the (light blue)
long-dashed is the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{-}\pi^{+}$ reflection, and the (blue) solid line is the total.

The invariant mass of the selected ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\pi^{+}\pi^{-}$ combinations, where the dimuon pair is constrained to
have the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ mass, is shown in Fig.
4. There are signal peaks at both the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ and $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ masses on top of the background.
Double-Gaussian functions are used to fit both signal peaks. They differ only
in their mean values, which are determined by the data. The core Gaussian
width is also allowed to vary, while the fraction and width ratio of the
second Gaussian is fixed to that obtained in the fit of $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\phi$ events. (The details of the fit are given in Ref. [10].) Other
components in the fit model take into account background contributions. One
source is from $B^{-}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{-}$decays, which contributes when the $K^{-}$ is misidentifed as a
$\pi^{-}$ and then combined with a random $\pi^{+}$; the smaller
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\pi^{-}$ mode contributes when it
is combined with a random $\pi^{+}$. The next source contains $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\eta^{\prime}(\rightarrow\rho\gamma)$ and $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\phi(\rightarrow\pi^{+}\pi^{-}\pi^{0})$ decays where the $\gamma$ and
the $\pi^{0}$ are ignored respectively. Finally there is a $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{-}\pi^{+}$ reflection where the $K^{-}$ is misidentified as
$\pi^{-}$. Here and elsewhere charged conjugated modes are included when
appropriate. The exponential combinatorial background shape is taken from
same-sign combinations, that are the sum of
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\pi^{+}\pi^{+}$ and
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\pi^{-}\pi^{-}$ candidates. The
shapes of the other components are taken from the simulation with their
normalizations allowed to vary. The fit gives $5287\pm 112$ signal and
$3212\pm 80$ background candidates within $\pm
20$$\mathrm{\,Me\kern-1.00006ptV}$ of the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ mass peak, where a
$K^{0}_{\rm\scriptscriptstyle S}$ veto, discussed later, is applied.

We use the well measured
$B^{-}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{-}$ mode as a
normalization channel to determine the branching fractions. To minimize the
systematic uncertainty from the BDT selection, we employ a similar selection
on $B^{-}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{-}$ decays
after requiring the same pre-selection except for particle identification
criteria on the $K^{-}$ candidates. Similar variables are used for the BDT
except that the variables describing the combination of $\pi^{+}$ and
$\pi^{-}$ in the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\pi^{+}\pi^{-}$
final state are replaced by ones describing the $K^{-}$ meson. For BDT
training, the signal sample uses simulated events and the background sample
consists of the data events in the region
$5400<m({J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{-})<5450$$\mathrm{\,Me\kern-1.00006ptV}$. The resulting invariant
mass distribution of the candidates satisfying BDT classifier $>0.05$ is shown
in Fig. 5. Fitting the distribution with a double-Gaussian function for the
signal and linear function for the background gives $350{,}727\pm 633$ signal
and $4756\pm 103$ background candidates within $\pm
20$$\mathrm{\,Me\kern-1.00006ptV}$ of the $B^{-}$ mass peak.

Figure 5: Invariant mass of ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{-}$ combinations. The data points are fitted with a double-Gaussian
function for signal and a linear function for background. The dotted line
shows the background, and the (blue) solid line is the total.

##  3 Analysis formalism

We apply a formalism similar to that used in Belle’s analysis [38] of $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow K^{-}\pi^{+}\chi_{c1}$
decays and later used in LHCb’s analysis of $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\pi^{+}\pi^{-}$ decays [10]. The decay $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\pi^{+}\pi^{-}$, with ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\rightarrow\mu^{+}\mu^{-}$, can be described by four variables. These
are taken to be the invariant mass squared of
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\pi^{+}$ ($s_{12}\equiv
m^{2}({J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\pi^{+})$), the invariant
mass squared of $\pi^{+}\pi^{-}$ ($s_{23}\equiv m^{2}(\pi^{+}\pi^{-})$), where
we use label 1 for ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$, 2 for
$\pi^{+}$ and 3 for $\pi^{-}$, the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}$ helicity angle ($\theta_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}$), which is the angle of the $\mu^{+}$ in the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ rest frame with respect to the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ direction in the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ rest frame, and the angle between
the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ and $\pi^{+}\pi^{-}$ decay
planes ($\chi$) in the $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ rest
frame. To improve the resolution of these variables we perform a kinematic fit
constraining the $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ and
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ masses to their nominal values
[31], and recompute the final state momenta. To simplify the probability
density function, we analyze the decay process after integrating over $\chi$,
which eliminates several interference terms.

###  3.1 The decay model for $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\pi^{+}\pi^{-}$

The overall probability density function (PDF) given by the sum of signal,
$S$, and background, $B$, functions is

$F(s_{12},s_{23},\theta_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}})=\frac{f_{\rm sig}}{{\cal{N}}_{\rm
sig}}\varepsilon(s_{12},s_{23},\theta_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}})S(s_{12},s_{23},\theta_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}})+\frac{(1-f_{\rm sig})}{{\cal{N}}_{\rm
bkg}}B(s_{12},s_{23},\theta_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}),$
(1)

where $f_{\rm sig}$ is the fraction of the signal in the fitted region and
$\varepsilon$ is the detection efficiency. The fraction of the signal is
obtained from the mass fit and is fixed for the subsequent analysis. The
normalization factors are given by

$\displaystyle{\cal{N}}_{\rm sig}$ $\displaystyle=$
$\displaystyle\int\\!\varepsilon(s_{12},s_{23},\theta_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}})S(s_{12},s_{23},\theta_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}})\,ds_{12}ds_{23}d\cos\theta_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}},$ $\displaystyle{\cal{N}}_{\rm bkg}$ $\displaystyle=$
$\displaystyle\int\\!B(s_{12},s_{23},\theta_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}})\,ds_{12}ds_{23}d\cos\theta_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}.$ (2)

The event distribution for $m^{2}(\pi^{+}\pi^{-})$ versus
$m^{2}({J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\pi^{+})$ in Fig. 6 shows
obvious structure in $m^{2}(\pi^{+}\pi^{-})$. To investigate if there are
visible exotic structures in the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\pi^{+}$ system as claimed in similar decays [39], we examine the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\pi^{+}$ mass distribution shown
in Fig. 7 (a). No resonant effects are evident. Figure 7 (b) shows the
$\pi^{+}\pi^{-}$ mass distribution. There is a clear peak at the $\rho(770)$
region, a small bump around 1250$\mathrm{\,Me\kern-1.00006ptV}$, but no
evidence for the $f_{0}(980)$ resonance. The favored $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{0}_{\rm\scriptscriptstyle S}$ decay is mostly rejected by the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ vertex $\chi^{2}$ selection, but
about 150 such events remain. We eliminate them by excluding the candidates
that have $|m(\pi^{+}\pi^{-})-m_{K^{0}_{\rm\scriptscriptstyle S}}|<$
25$\mathrm{\,Me\kern-1.00006ptV}$, where $m_{K^{0}_{\rm\scriptscriptstyle S}}$
is the $K^{0}_{\rm\scriptscriptstyle S}$ mass [31].

Figure 6: Distribution of $m^{2}(\pi^{+}\pi^{-})$ versus
$m^{2}({J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\pi^{+})$ for $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ candidate decays within $\pm
20$$\mathrm{\,Me\kern-1.00006ptV}$ of the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ mass.

Figure 7: Distribution of (a) $m({J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\pi^{+})$ and (b) $m(\pi^{+}\pi^{-})$ for $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\pi^{+}\pi^{-}$ candidate decays within $\pm
20$$\mathrm{\,Me\kern-1.00006ptV}$ of $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ mass shown with the solid line.
The (red) points with error bars show the background contribution determined
from $m({J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\pi^{+}\pi^{-})$ fits
performed in each bin.

####  3.1.1 The signal function

The signal function for $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ is
taken to be the coherent sum over resonant states that can decay into
$\pi^{+}\pi^{-}$, plus a possible non-resonant S-wave contribution333The
interference terms between different helicities are zero because we integrate
over the angular variable $\chi$.

$S(s_{12},s_{23},\theta_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}})=\sum_{\lambda=0,\pm
1}\left|\sum_{i}a^{R_{i}}_{\lambda}e^{i\phi^{R_{i}}_{\lambda}}\mathcal{A}_{\lambda}^{R_{i}}(s_{12},s_{23},\theta_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}})\right|^{2},$ (3)

where
$\mathcal{A}_{\lambda}^{R_{i}}(s_{12},s_{23},\theta_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}})$ is the amplitude of the decay via an intermediate resonance $R_{i}$
with helicity $\lambda$. Each $R_{i}$ has an associated amplitude strength
$a_{\lambda}^{R_{i}}$ for each helicity state $\lambda$ and a phase
$\phi_{\lambda}^{R_{i}}$. Note that the spin-0 component can only have a
$\lambda=0$ term. The amplitudes for each $i$ are defined as

$\mathcal{A}_{\lambda}^{R}(s_{12},s_{23},\theta_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}})=F_{B}^{(L_{B})}\;F_{R}^{(L_{R})}\;A_{R}(s_{23})\;\Big{(}\frac{P_{B}}{m_{B}}\Big{)}^{L_{B}}\;\Big{(}\frac{P_{R}}{\sqrt{s_{23}}}\Big{)}^{L_{R}}\;T_{\lambda}\;\Theta_{\lambda}(\theta_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}),$ (4)

where $P_{B}$ is the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ momentum
in the $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ rest frame and
$P_{R}$ is the momentum of either of the two pions in the dipion rest frame,
$m_{B}$ is the $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ mass,
$F_{B}^{(L_{B})}$ and $F_{R}^{(L_{R})}$ are the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ meson and $R$ resonance Blatt-
Weisskopf barrier factors [40], $L_{B}$ is the orbital angular momentum
between the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ and
$\pi^{+}\pi^{-}$ system, and $L_{R}$ is the orbital angular momentum in the
$\pi^{+}\pi^{-}$ decay and is equal to the spin of resonance $R$ because pions
have spin-0. Since the parent $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ has spin-0 and the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ is a vector, when the
$\pi^{+}\pi^{-}$ system forms a spin-0 resonance, $L_{B}=1$ and $L_{R}=0$. For
$\pi^{+}\pi^{-}$ resonances with non-zero spin, $L_{B}$ can be 0, 1 or 2 (1, 2
or 3) for $L_{R}=1(2)$ and so on. We take the lowest $L_{B}$ as the default
and consider the other possibilities in the systematic uncertainty.

The Blatt-Weisskopf barrier factors $F_{B}^{(L_{B})}$ and $F_{R}^{(L_{R})}$
are

$\displaystyle F^{(0)}$ $\displaystyle=$ $\displaystyle 1,$ $\displaystyle
F^{(1)}$ $\displaystyle=$ $\displaystyle\frac{\sqrt{1+z_{0}}}{\sqrt{1+z}},$
(5) $\displaystyle F^{(2)}$ $\displaystyle=$
$\displaystyle\frac{\sqrt{z_{0}^{2}+3z_{0}+9}}{\sqrt{z^{2}+3z+9}}.$

For the $B$ meson $z=r^{2}P_{B}^{2}$, where the hadron scale $r$ is taken as
$5.0\mathrm{\,Ge\kern-1.00006ptV}^{-1}$, and for the $R$ resonance
$z=r^{2}P_{R}^{2}$ with $r$ taken as $1.5\mathrm{\,Ge\kern-1.00006ptV}^{-1}$
[41]. In both cases $z_{0}=r^{2}P_{0}^{2}$ where $P_{0}$ is the decay daughter
momentum calculated at the resonance pole mass.

The angular term, $T_{\lambda}$, is obtained using the helicity formalism and
is defined as

$T_{\lambda}=d^{J}_{\lambda 0}(\theta_{\pi\pi}),$ (6)

where $d$ is the Wigner $d$-function, $J$ is the resonance spin,
$\theta_{\pi\pi}$ is the $\pi^{+}\pi^{-}$ resonance helicity angle which is
defined as the angle of the $\pi^{+}$ in the $\pi^{+}\pi^{-}$ rest frame with
respect to the $\pi^{+}\pi^{-}$direction in the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ rest frame and calculated from the
other variables as

$\cos\theta_{\pi\pi}=\frac{\left[m^{2}({J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\pi^{+})-m^{2}({J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\pi^{-})\right]m(\pi^{+}\pi^{-})}{4P_{R}P_{B}m_{B}}.$ (7)

The ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ helicity dependent term
$\Theta_{\lambda}(\theta_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}})$ is
defined as

$\displaystyle\Theta_{\lambda}(\theta_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}})$ $\displaystyle=$
$\displaystyle\sqrt{\sin^{2}\theta_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}}\;\;\;\;\;\;\;\;\;\text{for}\;\;\text{helicity}=0$ (8)
$\displaystyle=$
$\displaystyle\sqrt{\frac{1+\cos^{2}\theta_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}}{2}}\;\;\text{for}\;\;\text{helicity}=\pm 1.$

The function $A_{R}(s_{23})$ describes the mass squared shape of the resonance
$R$, that in most cases is a Breit-Wigner (BW) amplitude. Complications arise,
however, when a new decay channel opens close to the resonant mass. The
proximity of a second threshold distorts the line shape of the amplitude. This
happens for the $f_{0}(980)$ resonance because the $K^{+}K^{-}$ decay channel
opens. Here we use a Flatté model [42] which is described below.

The BW amplitude for a resonance decaying into two spin-0 particles, labeled
as 2 and 3, is

$A_{R}(s_{23})=\frac{1}{m^{2}_{R}-s_{23}-im_{R}\Gamma(s_{23})}~{},$ (9)

where $m_{R}$ is the resonance pole mass, $\Gamma(s_{23})$ is its energy-
dependent width that is parametrized as

$\Gamma(s_{23})=\Gamma_{0}\left(\frac{P_{R}}{P_{0}}\right)^{2L_{R}+1}\left(\frac{m_{R}}{\sqrt{s_{23}}}\right)F^{2}_{R}~{}.$
(10)

Here $\Gamma_{0}$ is the decay width when the invariant mass of the daughter
combinations is equal to $m_{R}$.

The Flatté model is parametrized as

$A_{R}(s_{23})=\frac{1}{m_{R}^{2}-s_{23}-im_{R}(g_{\pi\pi}\rho_{\pi\pi}+g_{KK}\rho_{KK})}.$
(11)

The constants $g_{\pi\pi}$ and $g_{KK}$ are the $f_{0}(980)$ couplings to
$\pi\pi$ and $K\kern 1.99997pt\overline{\kern-1.99997ptK}{}$ final states
respectively. The $\rho$ factors account for the Lorentz-invariant phase space
and are given as

$\displaystyle\rho_{\pi\pi}$ $\displaystyle=$
$\displaystyle\frac{2}{3}\sqrt{1-\frac{4m^{2}_{\pi^{\pm}}}{m^{2}(\pi^{+}\pi^{-})}}+\frac{1}{3}\sqrt{1-\frac{4m^{2}_{\pi^{0}}}{m^{2}(\pi^{+}\pi^{-})}},$
(12) $\displaystyle\rho_{KK}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\sqrt{1-\frac{4m^{2}_{K^{\pm}}}{m^{2}(\pi^{+}\pi^{-})}}+\frac{1}{2}\sqrt{1-\frac{4m^{2}_{K^{0}}}{m^{2}(\pi^{+}\pi^{-})}}.$
(13)

For non-resonant processes, the amplitude
$\mathcal{A}(s_{12},s_{23},\theta_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}})$ is derived from Eq. 4, considering that the $\pi^{+}\pi^{-}$ system
is S-wave (i.e. $L_{R}=0$, $L_{B}=1$) and $A_{R}(s_{23})$ is constant over the
phase space $s_{12}$ and $s_{23}$. Thus, it is parametrized as

$\mathcal{A}(s_{12},s_{23},\theta_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}})=\frac{P_{B}}{m_{B}}\sqrt{\sin^{2}\theta_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}}.$ (14)

####  3.1.2 Detection efficiency

Figure 8: Distributions of $\cos\theta_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}$ for the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\pi^{+}\pi^{-}$
simulated sample in (a) the entire dipion mass region and (b) $\rho(770)$
region.

The detection efficiency is determined from a sample of two million $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}(\rightarrow\mu^{+}\mu^{-})\pi^{+}\pi^{-}$ simulated events that are
generated uniformly in phase space. Both $s_{12}$ and $s_{13}$ are centered at
about $18.4\mathrm{\,Ge\kern-1.00006ptV}^{2}$. We model the detection
efficiency using the symmetric dimensionless Dalitz plot observables

$x=s_{12}/{\rm\mathrm{\,Ge\kern-1.00006ptV}}^{2}-18.4,~{}~{}~{}~{}{\rm
and}~{}~{}~{}~{}y=s_{13}/{\rm\mathrm{\,Ge\kern-1.00006ptV}}^{2}-18.4.$ (15)

These variables are related to $s_{23}$ since

$s_{12}+s_{13}+s_{23}=m^{2}_{B}+m^{2}_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}+m^{2}_{\pi^{+}}+m^{2}_{\pi^{-}}.$ (16)

The acceptance in $\cos\theta_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$
is not uniform, but depends on $s_{23}$, as shown in Fig. 8. If the efficiency
was independent of $s_{23}$, then the curves would have the same shape. On the
other hand, no clear dependence on $s_{12}$ is seen. Thus the efficiency model
can be expressed as

$\varepsilon(s_{12},s_{23},\theta_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}})=\varepsilon_{1}(x,y)\times\varepsilon_{2}(s_{23},\theta_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}).$ (17)

To study the $\cos\theta_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$
acceptance, we fit the $\cos\theta_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}$ distributions from simulation in 24 bins of $m^{2}(\pi^{+}\pi^{-})$
with the function

$\varepsilon_{2}(s_{23},\theta_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}})=\frac{1+a\cos^{2}\theta_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}}{2+2a/3},$ (18)

giving 24 values of $a$ as a function of $m^{2}(\pi^{+}\pi^{-})$. The
resultant distribution shown in Fig. 9 can be described by an exponential
function

$a(s_{23})=\exp(a_{1}+a_{2}s_{23}),$ (19)

with $a_{1}=-1.48\pm 0.20$ and $a_{2}=(-1.45\pm
0.33)\mathrm{\,Ge\kern-1.00006ptV}^{-2}$.

Figure 9: Exponential fit to the acceptance parameter $a(s_{12})$ used in Eq.
18.

Equation 18 is normalized with respect to
$\cos\theta_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$. Thus, after
integrating over $\cos\theta_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$,
Eq. 17 becomes

$\int_{-1}^{+1}\varepsilon(s_{12},s_{23},\theta_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}})d\cos\theta_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}=\varepsilon_{1}(x,y).$ (20)

This term of the efficiency is parametrized as a symmetric fourth order
polynomial function given by

$\displaystyle\varepsilon_{1}(x,y)$ $\displaystyle=$ $\displaystyle
1+\epsilon^{\prime}_{1}(x+y)+\epsilon^{\prime}_{2}(x+y)^{2}+\epsilon^{\prime}_{3}xy+\epsilon^{\prime}_{4}(x+y)^{3}+\epsilon^{\prime}_{5}xy(x+y)$
(21)
$\displaystyle+\epsilon^{\prime}_{6}(x+y)^{4}+\epsilon^{\prime}_{7}xy(x+y)^{2}+\epsilon^{\prime}_{8}x^{2}y^{2},$

where the $\epsilon^{\prime}_{i}$ are the fit parameters.

Table 1: Efficiency parameters to describe the acceptance on the signal Dalitz-plot. | Parameter | Value
---|---|---
| $\epsilon^{\prime}_{1}$ |  $0.142\pm 0.010$
| $\epsilon^{\prime}_{2}$ |  $0.101\pm 0.014$
| $\epsilon^{\prime}_{3}$ |  $0.0082\pm 0.0005$
| $\epsilon^{\prime}_{4}$ |  $0.027\pm 0.007$
| $\epsilon^{\prime}_{5}$ |  $0.0052\pm 0.0003$
| $\epsilon^{\prime}_{6}$ |  $0.0028\pm 0.0010$
| $\epsilon^{\prime}_{7}$ |  $0.00074\pm 0.00017$
| $\epsilon^{\prime}_{8}$ | $-0.000105\pm 0.000008$
| $\chi^{2}/\rm ndf$ | 308/298

Figure 10 shows the polynomial function

Figure 10: Parametrized detection efficiency as a function of
$m^{2}(\pi^{+}\pi^{-})$ versus $m^{2}({J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\pi^{+})$ determined from simulation. The $z$-axis scale is arbitrary.

obtained from a fit to the Dalitz-plot distributions of simulated events. The
projections of the fit are shown in Fig. 11 and the resulting parameters are
given in Table 1.

Figure 11: Projections onto (a) $m^{2}({J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\pi^{+})$ and (b) $m^{2}(\pi^{+}\pi^{-})$ of the simulated Dalitz plot
used to determine the efficiency parameters. The points represent the
simulated event distributions and the curves the projections of the polynomial
fits.

####  3.1.3 Background composition

Backgrounds from $B$ decays into ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}$ final states have already been discussed in Section 2. The main
background source is combinatorial and its shape can be determined from the
same-sign $\pi^{\pm}\pi^{\pm}$ combinations within $\pm
20$$\mathrm{\,Me\kern-1.00006ptV}$ of the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ mass peak; this region also
contains the small $B^{-}$ background. In addition, there is background
arising from partially reconstructed $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ decays including $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\eta^{\prime}(\rightarrow\rho\gamma)$, $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\phi(\rightarrow\pi^{+}\pi^{-}\pi^{0})$, and a $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{-}\pi^{+}$ reflection, which cannot be present in same-sign
combinations. We use simulated samples of inclusive $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ decays, and exclusive $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\kern 1.99997pt\overline{\kern-1.99997ptK}{}^{*0}(892)$ and $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\kern 1.99997pt\overline{\kern-1.99997ptK}{}^{*0}_{2}(1430)$ decays to
model the additional backgrounds. The background fraction of each source is
studied by fitting the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\pi^{+}\pi^{-}$ candidate invariant mass distributions in bins of
$m^{2}(\pi^{+}\pi^{-})$. The resulting background distribution in the $\pm
20$$\mathrm{\,Me\kern-1.00006ptV}$ $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ signal region is shown in Fig. 12.
It is fit by histograms from the same-sign combinations and two additional
simulations, giving a partially reconstructed $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ background of 12.8%, and a
reflection background that is 5.2% of the total background.

Figure 12: The $m^{2}(\pi\pi)$ distribution of background. The (black)
histogram with error bars shows the same-sign data combinations with
additional background from simulation, the (blue) points with error bars show
the background obtained from the mass fits, the (black) dashed line is the
partially reconstructed $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$
background, and the (red) dotted is the misidentified $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{-}\pi^{+}$ contribution.

The background is parametrized as

$B(s_{12},s_{23},\theta_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}})=\frac{m(\pi^{+}\pi^{-})}{2P_{R}P_{B}m_{B}}B_{1}(s_{23},\cos\theta_{\pi\pi})\times\left(1+\alpha\cos^{2}\theta_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}\right),$ (22)

where the first part $\frac{m(\pi^{+}\pi^{-})}{2P_{R}P_{B}m_{B}}$ converts
phase space from $s_{12}$ to $\cos\theta_{\pi\pi}$, and

$\displaystyle B_{1}(s_{23},\cos\theta_{\pi\pi})=$
$\displaystyle\left[B_{2}(\zeta)\frac{p_{B}}{m_{B}}+\frac{b_{0}}{(m^{2}_{0}-s_{23})^{2}+m_{0}^{2}\Gamma_{0}^{2}}\right]$
(23)
$\displaystyle\times\frac{1+c_{1}q(\zeta)|\cos\theta_{\pi\pi}|+c_{4}p(\zeta)\cos^{2}\theta_{\pi\pi}}{2[1+c_{1}q(\zeta)/2+c_{4}p(\zeta)/3]}.$

The variable $\zeta=2(s_{23}-s_{\rm min})/(s_{\rm max}-s_{\rm min})-1$, where
$s_{\rm min}$ and $s_{\rm max}$ give the fit boundaries, $B_{2}(\zeta)$ is a
fifth-order Chebychev polynomial with parameters $b_{i}$ ($i=1$–5), and
$q(\zeta)$ and $p(\zeta)$ are both second-order Chebychev polynomials with
parameters $c_{i}$ ($i$=2, 3, 5, 6), and $c_{1}$, and $c_{4}$ are free
parameters. In order to better approximate the real background in the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ signal region, the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\pi^{\pm}\pi^{\mp}$ candidates
are kinematically constrained to the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ mass. A fit to the same-sign
sample, with additional background from simulation, determines $b_{i}$,
$c_{i}$, $m_{0}$ and $\Gamma_{0}$. Figure 13 shows the mass squared
projections from the fit. The fitted background parameters are shown in Table
2.

The $\left(1+\alpha\cos^{2}\theta_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}\right)$ term is a function of the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ helicity angle. The
$\cos\theta_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$ distribution of
background is shown in Fig. 14, and is fit with the function
$1+\alpha\cos^{2}\theta_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$ that
determines the parameter $\alpha=-0.38\pm 0.04$. We have verified that
$\alpha$ is independent of $s_{23}$.

Figure 13: Projections of invariant mass squared of (a) $m^{2}({J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\pi^{+})$ and (b) $m^{2}(\pi^{+}\pi^{-})$ of the background Dalitz plot. The points with error bars show the same-sign combinations with additional background from simulation. Figure 14: distribution of the background in $\cos\theta_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$ resulting from ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\pi^{+}\pi^{-}$ candidate mass fits in each bin of $\cos\theta_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$. The curve represents the fitted function $1+\alpha\cos^{2}\theta_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$. Table 2: Parameters for the background model used in Eq. 23. | Parameter | Value
---|---|---
| $b_{0}$ | $(4.4\pm 1.2)\times 10^{-3}\mathrm{\,Ge\kern-1.00006ptV}^{4}$
| $m_{0}$ |  $0.767\pm 0.005\mathrm{\,Ge\kern-1.00006ptV}$
| $\Gamma_{0}$ |  $0.101\pm 0.015\mathrm{\,Ge\kern-1.00006ptV}$
| $b_{1}$ | $-0.52\pm 0.07$
| $b_{2}$ | $0.22\pm 0.05$
| $b_{3}$ | $-0.14\pm 0.06$
| $b_{4}$ | $0.11\pm 0.04$
| $b_{5}$ | $-0.06\pm 0.04$
| $c_{1}$ | $-0.70\pm 0.04$
| $c_{2}$ | $-0.4\pm 0.3$
| $c_{3}$ | $1.9\pm 0.2$
| $c_{4}$ | $0.42\pm 0.03$
| $c_{5}$ | $1.7\pm 0.8$
| $c_{6}$ | $2.5\pm 0.8$
| $\chi^{2}/{\rm ndf}$ | 252/284

###  3.2 Fit fractions

While a complete description of the decay is given in terms of the fitted
amplitudes and phases, the knowledge of the contribution of each component can
be summarized by defining a fit fraction, ${\cal{F}}^{R}_{\lambda}$, as the
integration of the squared amplitude of $R$ over the Dalitz plot divided by
the integration of the entire signal function,

${\cal{F}}^{R}_{\lambda}=\frac{{\int}\left|a^{R}_{\lambda}e^{i\phi^{R}_{\lambda}}\mathcal{A}_{\lambda}^{R}(s_{12},s_{23},\theta_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}})\right|^{2}ds_{12}\;ds_{23}\;d\cos\theta_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}}{{\int}S(s_{12},s_{23},\theta_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}})~{}ds_{12}\;ds_{23}\;d\cos\theta_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}}.$ (24)

Note that the sum of the fit fractions over all $\lambda$ and $R$ is not
necessarily unity due to the potential presence of interference between two
resonances. If the Dalitz plot has more destructive interference than
constructive interference, the total fit fraction will be greater than one.
Interference term fractions are given by

${\cal{F}}^{RR^{\prime}}_{\lambda}=\mathcal{R}e\left(\frac{{\int}a^{R}_{\lambda}\;a^{R^{\prime}}_{\lambda}e^{i(\phi^{R}_{\lambda}-\phi^{R^{\prime}}_{\lambda})}\mathcal{A}_{\lambda}^{R}(s_{12},s_{23},\theta_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}){\mathcal{A}_{\lambda}^{R^{\prime}}}^{*}(s_{12},s_{23},\theta_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}})ds_{12}\;ds_{23}\;d\cos\theta_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}}{{\int}S(s_{12},s_{23},\theta_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}})~{}ds_{12}\;ds_{23}\;d\cos\theta_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}}\right),$ (25)

and the sum of the two is

$\sum_{\lambda}\left(\sum_{R}{\cal{F}}^{R}_{\lambda}+\sum_{RR^{\prime}}^{R\neq
R^{\prime}}{\cal{F}}^{RR^{\prime}}_{\lambda}\right)=1.$ (26)

Note that interference terms between different spin-$J$ states vanish, because
the $d^{J}_{\lambda 0}$ angular functions in $\mathcal{A}^{R}_{\lambda}$ are
orthogonal.

The statistical errors of the fit fractions depend on the statistical errors
of every fitted magnitude and phase, and their correlations. Therefore, to
determine the uncertainties the covariance matrix and parameter values from
the fit are used to generate 500 sample parameter sets. For each set, the fit
fractions are calculated. The distributions of the obtained fit fractions are
described by bifurcated Gaussian functions. The widths of the Gaussians are
taken as the statistical errors on the corresponding parameters. The
correlations of fitted parameters are also taken into account.

##  4 Final state composition

###  4.1 Resonance models

To study the resonant structures of the decay $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\pi^{+}\pi^{-}$ we use those combinations with an invariant mass within
$\pm 20$$\mathrm{\,Me\kern-1.00006ptV}$ of the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ mass peak and apply a
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{0}_{\rm\scriptscriptstyle S}$
veto. The total number of remaining candidates is 8483, of which $3212\pm 80$
are attributed to background. Possible resonances in the decay $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\pi^{+}\pi^{-}$ are listed in Table 3. In addition, there could be some
contribution from non-resonant $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\pi^{+}\pi^{-}$ decays.

Table 3: Possible resonances in the $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\pi^{+}\pi^{-}$ decay mode. Resonance | Spin | Helicity | Resonance
---|---|---|---
|  |  | formalism
$f_{0}(500)$ | 0 | 0 | BW
$\rho(770)$ | 1 | $0,\pm 1$ | BW
$\omega(782)$ | 1 | $0,\pm 1$ | BW
$f_{0}(980)$ | 0 | 0 | Flatté
$f_{2}(1270)$ | 2 | $0,\pm 1$ | BW
$f_{0}(1370)$ | 0 | 0 | BW
$\rho(1450)$ | 1 | $0,\pm 1$ | BW
$f_{0}(1500)$ | 0 | 0 | BW
$\rho(1700)$ | 1 | $0,\pm 1$ | BW
$f_{0}(1710)$ | 0 | 0 | BW
Table 4: Breit-Wigner resonance parameters. Resonance | Mass ($\mathrm{\,Me\kern-1.00006ptV}$) | Width ($\mathrm{\,Me\kern-1.00006ptV}$) | Source
---|---|---|---
$f_{0}(500)$ |  $513\pm 32$ |  $335\pm 67$ | CLEO [43]
$\rho(770)$ |  $775.49\pm 0.34$ | $149.1\pm 0.8$ | PDG [31]
$\omega(782)$ |  $782.65\pm 0.12$ |  $8.49\pm 0.08$ | PDG [31]
$f_{2}(1270)$ | $1275.1\pm 1.2$ | $185.1_{-2.4}^{+2.9}$ | PDG [31]
$f_{0}(1370)$ | $1475\pm 6$ |  $113\pm 11$ |  LHCb [10]
$\rho(1450)$ |  $1465\pm 25$ |  $400\pm 60$ | PDG [31]
$f_{0}(1500)$ | $1505\pm 6$ |  $109\pm 7$ | PDG [31]
$\rho(1700)$ |  $1700\pm 20$ |  $250\pm 100$ | PDG [31]
$f_{0}(1710)$ | $1720\pm 6$ |  $135\pm 8$ | PDG [31]

The masses and widths of the BW resonances are listed in Table 4. When used in
the fit they are fixed to these values except for the parameters of the
$f_{0}(500)$ resonance which are constrained by their uncertainties. Besides
the mass and width, the Flatté resonance shape has two additional parameters
$g_{\pi\pi}$ and $g_{KK}$, which are also fixed in the fit to values obtained
in our previous Dalitz analysis of $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\pi^{+}\pi^{-}$ [10], where a large fraction of $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ decays are to
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}f_{0}(980)$. The parameters are
taken to be $m_{0}=939.9\pm 6.3$$\mathrm{\,Me\kern-1.00006ptV}$,
$g_{\pi\pi}=199\pm 30$$\mathrm{\,Me\kern-1.00006ptV}$ and
$g_{KK}/g_{\pi\pi}=3.0\pm 0.3$. All background and efficiency parameters are
fixed in the fit.

To determine the complex amplitudes in a specific model, the data are fitted
maximizing the unbinned likelihood given as

$\mathcal{L}=\prod_{i=1}^{N}F(s^{i}_{12},s^{i}_{23},\theta^{i}_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}),$ (27)

where $N$ is the total number of candidates, and $F$ is the total PDF defined
in Eq. 1. The PDF is constructed from the signal fraction $f_{\rm sig}$, the
efficiency model
$\varepsilon(s_{12},s_{23},\theta_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}})$, the background model
$B(s_{12},s_{23},\theta_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}})$, and
the signal model
$S(s_{12},s_{23},\theta_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}})$. In
order to ensure proper convergence using the maximum likelihood method, the
PDF needs to be normalized. This is accomplished by first normalizing the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ helicity dependent part
$\varepsilon(s_{23},\theta_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}})\Theta_{\lambda}(\theta_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}})$ over $\cos\theta_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$ by
analytical integration. This integration results in additional factors as a
function of $s_{23}$. We then normalize the mass dependent part multiplied by
the additional factors using numerical integration over 500$\times$500 bins.

The fit determines the relative amplitude magnitudes $a_{\lambda}^{R_{i}}$ and
phases $\phi_{\lambda}^{R_{i}}$ defined in Eq. 3; we choose to fix
$a_{0}^{\rho(770)}$ to 1. As only relative phases are physically meaningful,
one phase in each helicity grouping has to be fixed; we choose to fix those of
the $f_{0}(500)$ and the $\rho(770)$ ($|\lambda|=1$) to 0. In addition, since
the final state ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\pi^{+}\pi^{-}$
is a self-charge-conjugate mode and as we do not determine the $B$ flavor, the
signal function is an average of $B^{0}$ and $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ decays. If we do not consider
$\pi^{+}\pi^{-}$ partial waves of a higher order than D-wave, then we can
express the differential decay rate derived from Eqs. 3, 4 and 8 in terms of
S-, P-, and D-waves including helicity 0 and $\pm 1$

$\displaystyle\frac{d\overline{\Gamma}}{dm_{\pi\pi}d\cos\theta_{\pi\pi}d\cos\theta_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}}$ (28) $\displaystyle=$ $\displaystyle\left|{\cal
A}^{s}_{S_{0}}e^{i\phi^{s}_{S_{0}}}+{\cal
A}^{s}_{P_{0}}e^{i\phi^{s}_{P_{0}}}\cos\theta_{\pi\pi}+{\cal
A}^{s}_{D_{0}}e^{i\phi^{s}_{D_{0}}}\left(\frac{3}{2}\cos^{2}\theta_{\pi\pi}-\frac{1}{2}\right)\right|^{2}\sin^{2}\theta_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}$ $\displaystyle+$ $\displaystyle\left|{\cal A}^{s}_{P_{\pm
1}}e^{i\phi^{s}_{P_{\pm 1}}}\frac{1}{2}\sin\theta_{\pi\pi}+{\cal
A}^{s}_{D_{\pm 1}}e^{i\phi^{s}_{D_{\pm
1}}}\sqrt{\frac{3}{2}}\sin\theta_{\pi\pi}\cos\theta_{\pi\pi}\right|^{2}\frac{1+\cos^{2}\theta_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}}{2}$

for $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ decays, where ${\cal
A}^{s}_{k_{\lambda}}$ and $\phi^{s}_{k_{\lambda}}$ are the sum of amplitudes
and reference phase for the spin-$k$ resonance group, respectively. The
$B^{0}$ function for decays is similar, but $\theta_{\pi^{+}\pi^{-}}$ and
$\theta_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$ are changed to
$\pi-\theta_{\pi^{+}\pi^{-}}$ and
$\pi-\theta_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$ respectively, as a
result of using $\pi^{-}$ and $\mu^{-}$ to define the helicity angles,
yielding

$\displaystyle\frac{d\Gamma}{dm_{\pi\pi}d\cos\theta_{\pi\pi}d\cos\theta_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}}$ (29) $\displaystyle=$ $\displaystyle\left|{\cal
A}^{s}_{S_{0}}e^{i\phi^{s}_{S_{0}}}-{\cal
A}^{s}_{P_{0}}e^{i\phi^{s}_{P_{0}}}\cos\theta_{\pi\pi}+{\cal
A}^{s}_{D_{0}}e^{i\phi^{s}_{D_{0}}}\left(\frac{3}{2}\cos^{2}\theta_{\pi\pi}-\frac{1}{2}\right)\right|^{2}\sin^{2}\theta_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}$ $\displaystyle+$ $\displaystyle\left|{\cal A}^{s}_{P_{\pm
1}}e^{i\phi^{s}_{P_{\pm 1}}}\frac{1}{2}\sin\theta_{\pi\pi}-{\cal
A}^{s}_{D_{\pm 1}}e^{i\phi^{s}_{D_{\pm
1}}}\sqrt{\frac{3}{2}}\sin\theta_{\pi\pi}\cos\theta_{\pi\pi}\right|^{2}\frac{1+\cos^{2}\theta_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}}{2}.$

Summing Eqs. 28 and 29 results in cancellation of the interference involving
the $\lambda=0$ terms for spin-1, and the $\lambda=\pm 1$ terms for spin-2, as
they appear with opposite signs for $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ and $B^{0}$ decays. Therefore, we
have to fix one phase in spin-1 ($\lambda=0$) group ($\phi^{s}_{P_{0}}$) and
one in spin-2 ($\lambda=\pm 1$) group ($\phi^{s}_{D_{\pm 1}}$); the phases of
$\rho(770)$ ($\lambda=0$) and $f_{2}(1270)$ ($\lambda=\pm 1$) are fixed to
zero. The other phases in each corresponding group are relative to that of the
fixed resonance.

###  4.2 Fit results

To find the best model, we proceed by fitting with all the possible resonances
and a non-resonance (NR) component, then subsequently remove the most
insignificant component one at a time. We repeat this procedure until each
remaining contribution has more than 3 statistical standard deviation
($\sigma$) significance. The significance is estimated from the fit fraction
divided by its statistical uncertainty. The best fit model contains six
resonances, the $f_{0}(500)$, $f_{0}(980)$, $f_{2}(1270)$, $\rho(770)$,
$\rho(1450)$, and $\omega(782)$.

In order to compare the different models quantitatively an estimate of the
goodness of fit is calculated from three-dimensional partitions of the one
angular and two mass squared variables. We use the Poisson likelihood
$\chi^{2}$ [44] defined as

$\chi^{2}=2\sum_{i=1}^{N_{\rm
bin}}\left[x_{i}-n_{i}+n_{i}\text{ln}\left(\frac{n_{i}}{x_{i}}\right)\right],$
(30)

where $n_{i}$ is the number of events in the three dimensional bin $i$ and
$x_{i}$ is the expected number of events in that bin according to the fitted
likelihood function. A total of 1021 bins ($N_{\rm bin}$) are used to
calculate the $\chi^{2}$, based on the variables
$m^{2}({J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\pi^{+})$,
$m^{2}(\pi^{+}\pi^{-})$, and
$\cos\theta_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$. The
$\chi^{2}/\text{ndf}$ and the negative of the logarithm of the likelihood,
$\rm-ln\mathcal{L}$, of the fits are given in Table 5; ndf is equal to $N_{\rm
bin}-1-N_{\rm par}$, where $N_{\rm par}$ is the number of fitting parameters.
The difference between the best fit results and fits with one additional
component is taken as a systematic uncertainty. Figure 15 shows the best fit
model projections of $m^{2}(\pi^{+}\pi^{-})$,
$m^{2}({J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\pi^{+})$,
$\cos\theta_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$ and
$m(\pi^{+}\pi^{-})$. We calculate the fit fraction of each component using Eq.
24. For a P- or D-wave resonance, we report its total fit fraction by summing
all the helicity components, and the fraction of the helicity $\lambda=0$
component. The results are listed in Table 6. Systematic uncertainties will be
discussed in Section 6. Two interesting ratios of fit fractions are
($0.93_{-0.22-0.23}^{+0.37+0.47}$)% for $\omega(782)$ to $\rho(770)$, and
($9.5_{-3.4}^{+6.7}\pm 3.0$)% for $f_{0}(980)$ to $f_{0}(500)$.

The fit fractions of the interference terms are computed using Eq. 25 and
listed in Table 7. Table 8 shows the resonant phases from the best fit. For
the systematic uncertainty study, Table 9 shows the fit fractions of
components for the best model with one additional resonance.

Table 5: Values of $\chi^{2}/\text{ndf}$ and $\rm-ln\mathcal{L}$ of different resonance models. Resonance model | $\rm-ln\mathcal{L}$ | $\chi^{2}/\text{ndf}$ | Probability (%)
---|---|---|---
Best Model | 35292 | 1058/1003 | 11.1
Best Model + $\rho(1700)$ | 35284 | 1045/ 999 | 15.0
Best Model + NR | 35284 | 1058/1001 | 10.3
Best Model + $f_{0}(1370)$ | 35285 | 1047/1001 | 15.2
Best Model + $f_{0}(1500)$ | 35287 | 1049/1001 | 14.4
Best Model + $f_{0}(1710)$ | 35289 | 1052/1001 | 12.6

Figure 15: Dalitz fit projections of (a) $m^{2}(\pi^{+}\pi^{-})$, (b) $m^{2}({J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\pi^{+})$, (c) $\cos\theta_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$ and (d) $m(\pi^{+}\pi^{-})$ for the best model. The points with error bars are data, the signal fit is shown with a (red) dashed line, the background with a (black) dotted line, and the (blue) solid line represents the total. In (a) and (d), the shape variations near the $\rho(770)$ mass is due to $\rho(770)-\omega(782)$ interference, and the dip at the $K^{0}_{\rm\scriptscriptstyle S}$ mass [31] is due to the $K^{0}_{\rm\scriptscriptstyle S}$ veto. Table 6: Fit fractions and significances of contributing components for the best model, as well as the fractions of the helicity $\lambda=0$ part. The significance takes into account both statistical and systematic uncertainties. Components | Fit fraction (%) |  $\lambda=0$ fraction | Significance ($\sigma$)
---|---|---|---
$\rho(770)$ | $62.8_{-2.9-4.8}^{+4.8+2.8}$ | $0.63\pm 0.04_{-0.03}^{+0.06}$ | 11.2
$\omega(782)$ | $0.59_{-0.13-0.14}^{+0.23+0.27}$ | $0.30^{+0.26}_{-0.18}\pm 0.05$ | 3.1
$f_{0}(980)$ | $1.53^{+0.77+0.43}_{-0.50-0.35}$ | 1 | 2.5
$f_{2}(1270)$ | $8.9\pm 1.1\pm 1.0$ | $0.76\pm 0.06\pm 0.05$ | 5.9
$\rho(1450)$ | $5.3^{+2.5+5.6}_{-1.4-0.9}$ | $0.28^{+0.17+0.08}_{-0.13-0.12}$ | 3.2
$f_{0}(500)$ | $16.2\pm 2.0^{+6.0}_{-2.0}$ | 1 | 5.7
Sum | 95.2 |  |
Table 7: Interference fractions ${\cal{F}}_{\lambda}^{RR^{\prime}}$ (%) computed using Eq 25. Note that the diagonal elements are fit fractions defined in Eq 24.  |  | $\rho$ | $\omega$ | $\rho$ | $f_{0}$ | $f_{0}$ | $f_{2}$ | $\rho$ | $\omega$ | $\rho$ | $f_{2}$
---|---|---|---|---|---|---|---|---|---|---|---
|  | 770 | 782 | 1450 | 980 | 500 | 1270 | 770 | 782 | 1450 | 1270
| $|\lambda|$ | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1
$\rho(770)$ | 0 | 39.44 | $-0.02$ | $-0.89$ | 0 | 0 | 0 | 0 | 0 | 0 | 0
$\omega(782)$ | 0 |  |  0.18 | $-0.05$ | 0 | 0 | 0 | 0 | 0 | 0 | 0
$\rho(1450)$ | 0 |  |  |  1.47 | 0 | 0 | 0 | 0 | 0 | 0 | 0
$f_{0}(980)$ | 0 |  |  |  | 1.53 |  2.08 | 0 | 0 | 0 | 0 | 0
$f_{0}(500)$ | 0 |  |  |  |  | 16.15 | 0 | 0 | 0 | 0 | 0
$f_{2}(1270)$ | 0 |  |  |  |  |  | 6.72 | 0 | 0 | 0 | 0
$\rho(770)$ | 1 |  |  |  |  |  |  | 23.32 | 0.29 | 0 | 0
$\omega(782)$ | 1 |  |  |  |  |  |  |  | 0.41 | $-0.07$ | 0
$\rho(1450)$ | 1 |  |  |  |  |  |  |  |  |  3.80 | 0
$f_{2}(1270)$ | 1 |  |  |  |  |  |  |  |  |  | 2.14
Table 8: Resonant phases from the best fit. Components | Phase (deg)
---|---
$\rho(770)$, $\lambda=0$ | 0 (fixed)
$\rho(770)$, $|\lambda|=1$ | 0 (fixed)
$\omega(782)$, $\lambda=0$ | $-84\pm 31$
$\omega(782)$, $|\lambda|=1$ | $-70\pm 16$
$f_{0}(980)$ | $103\pm 17$
$f_{2}(1270)$, $\lambda=0$ | $-87\pm 12$
$f_{2}(1270)$, $|\lambda|=1$ | 0 (fixed)
$\rho(1450)$, $\lambda=0$ | $-162\pm 22$
$\rho(1450)$, $|\lambda|=1$ | $160\pm 48$
$f_{0}(500)$ | 0 (fixed)
Table 9: Fit fractions (%) of contributing components for the best model with adding one additional resonance. | Best | $+\rho(1700)$ | $+f_{0}(1370)$ | $+f_{0}(1500)$ | $+f_{0}(1710)$ | +NR
---|---|---|---|---|---|---
$\rho(770)$ | $62.8_{-2.9}^{+4.8}$ | $59.5_{-2.8}^{+3.1}$ | $62.6_{-2.5}^{+3.9}$ | $62.4_{-2.5}^{+4.1}$ | $63.3_{-3.0}^{+5.6}$ | $63.4_{-2.7}^{+3.8}$
$\omega(782)$ | $0.59_{-0.13}^{+0.23}$ | $0.58_{-0.15}^{+0.22}$ | $0.60_{-0.15}^{+0.26}$ | $0.60_{-0.15}^{+0.25}$ | $0.59_{-0.15}^{+0.25}$ | $0.59_{-0.14}^{+0.25}$
$f_{0}(980)$ | $1.53_{-0.50}^{+0.77}$ | $1.54_{-0.53}^{+0.75}$ | $1.54_{-0.55}^{+0.76}$ | $1.50_{-0.54}^{+0.78}$ | $1.55_{-0.51}^{+0.76}$ | $1.74_{-0.55}^{+0.80}$
$f_{2}(1270)$ | $8.9\pm 1.1$ | $8.1\pm 1.2$ | $8.8\pm 1.1$ | $8.8\pm 1.1$ | $8.9\pm 1.1$ | $8.8\pm 1.1$
$\rho(1450)$ | $5.3_{-1.4}^{+2.5}$ | $10.8_{-3.6}^{+5.4}$ | $4.7_{-1.1}^{+1.6}$ | $4.9_{-1.2}^{+1.9}$ | $5.7_{-2.3}^{+4.0}$ | $4.6_{-1.3}^{+2.1}$
$f_{0}(500)$ | $16.2\pm 2.0$ | $15.6\pm 1.9$ | $16.6\pm 2.0$ | $16.9\pm 2.1$ | $16.3\pm 2.1$ | $21.9\pm 3.8$
$\rho(1700)$ | - | $3.4_{-1.5}^{+2.7}$ | - | - | - | -
$f_{0}(1370)$ | - | - | $1.3_{-0.5}^{+0.8}$ | - | - | -
$f_{0}(1500)$ | - | - | - | $1.0_{-0.4}^{+0.7}$ | - | -
$f_{0}(1710)$ | - | - | - | - | $0.4_{-0.2}^{+0.4}$ | -
NR | - | - | - | - | - | $4.5_{-1.7}^{+2.7}$
Sum | 95.2 | 99.6 | 96.2 | 96.0 | 96.7 | 105.5

###  4.3 Helicity angle distributions

We show the helicity angle distributions in the $\rho(770)$ mass region
defined within one full width of the $\rho(770)$ resonance (the width values
are given in Table 4) in Fig. 16. The
$\cos\theta_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$ and
$\cos\theta_{\pi\pi}$ background subtracted and efficiency corrected
distributions for this mass region are presented in Fig. 17. The distributions
are in good agreement with the best fit model.

Figure 16: Helicity angle distributions of (a)
$\cos\theta_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$ ($\chi^{2}$/ndf
=15/20) and (b) $\cos\theta_{\pi\pi}$ ($\chi^{2}$/ndf =14/20) in the
$\rho(770)$ mass region defined within one full width of the $\rho(770)$ mass.
The points with error bars are data, the signal fit to the best model is shown
with a (red) dashed line, the background with a (black) dotted line, and the
(blue) solid line represents the total.

Figure 17: Background subtracted and efficiency corrected helicity
distributions of (a) $\cos\theta_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}$ ($\chi^{2}$/ndf =20/20) and (b) $\cos\theta_{\pi\pi}$ ($\chi^{2}$/ndf
=13/20) in the $\rho(770)$ mass region defined within one full width of the
$\rho(770)$ mass. The points with error bars are data and the solid blue lines
show the fit to the best model.

##  5 Branching fractions

Branching fractions are measured by normalizing to the well measured decay
mode $B^{-}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{-}$,
which has two muons in the final state and has the same triggers as the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\pi^{+}\pi^{-}$ decays. Assuming equal production of charged and neutral
$B$ mesons at the LHC due to isospin symmetry, the branching fraction is
calculated as

${\cal B}(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\pi^{+}\pi^{-})=\frac{N_{\kern
1.25995pt\overline{\kern-1.25995ptB}{}^{0}}/\epsilon_{\kern
1.25995pt\overline{\kern-1.25995ptB}{}^{0}}}{N_{B^{-}}/\epsilon_{B^{-}}}\times{\cal
B}(B^{-}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{-}),$ (31)

where $N$ and $\epsilon$ denote the yield and total efficiency of the decay of
interest. The branching fraction ${\cal
B}(B^{-}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{-})=(10.18\pm 0.42)\times 10^{-4}$ is determined from an average of
recent Belle [45] and BaBar [46] measurements that are corrected with respect
to the reported values, which assume equal production of charged and neutral
$B$ mesons at the $\Upsilon(4S)$, using the measured value of
$\frac{\Gamma(B^{+}B^{-})}{\Gamma(B^{0}\kern
1.25995pt\overline{\kern-1.25995ptB}{}^{0})}=1.055\pm 0.025$ [47].

Signal efficiencies are derived from simulations including trigger,
reconstruction, and event selection components. Since the efficiency to detect
the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\pi^{+}\pi^{-}$ final state
is not uniform across the Dalitz plane, the efficiency is averaged according
to the Dalitz model, where the best fit model is used. The
$K^{0}_{\rm\scriptscriptstyle S}$ veto efficiency is also taken into account.
Small corrections are applied to account for differences between the
simulation and the data. We measure the kaon and pion identification
efficiencies with respect to the simulation using
$D^{*+}\rightarrow\pi^{+}D^{0}(\rightarrow K^{-}\pi^{+})$ events selected from
data. The efficiencies are measured in bins of $p_{\rm T}$ and $\eta$ and the
averages are weighted using the signal event distributions in the data.
Furthermore, to ensure that the $p$ and $p_{\rm T}$ distributions of the
generated $B$ mesons are correct we weight the $B^{-}$ and $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ simulation samples using
$B^{-}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{-}$ and $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\kern 1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$ data, respectively.
Finally, the simulation samples are weighted with the charged tracking
efficiency ratio between data and simulation in bins of $p$ and $p_{\rm T}$ of
the track. The average of the weights is the correction factor. The total
correction factors are below 1.04 and largely cancel between the signal and
normalization channels. Multiplying the simulation efficiencies and correction
factors gives the total efficiency ($1.163\pm 0.003\pm 0.017$)% for $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\pi^{+}\pi^{-}$ and ($3.092\pm 0.012\pm 0.038$)% for
$B^{-}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{-}$, where the
first uncertainty is statistical and the second is systematic.

Using $N_{B^{-}}=350{,}727\pm 633$ and $N_{\kern
1.25995pt\overline{\kern-1.25995ptB}{}^{0}}=5287\pm 112$, we measure

${\cal B}(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\pi^{+}\pi^{-})=(3.97\pm 0.09\pm 0.11\pm 0.16)\times 10^{-5},$

where the first uncertainty is statistical, the second is systematic and the
third is due to the uncertainty of ${\cal
B}(B^{-}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{-})$. The
systematic uncertainties are discussed in Section 6. Our measured value is
consistent with and more precise than the previous BaBar measurement of
$(4.6\pm 0.7\pm 0.6)\times 10^{-5}$ [26].

Table 10 shows the branching fractions of resonant modes calculated by
multiplying the fit fraction and the total branching fraction of $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\pi^{+}\pi^{-}$. Since the $f_{0}(980)$ contribution has a significance
of less than 3$\sigma$ we quote also an upper limit of ${\cal{B}}\left(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}f_{0}(980)\right)\times{\cal{B}}\left(f_{0}(980)\rightarrow\pi^{+}\pi^{-}\right)<1.1\times
10^{-6}$ at 90% confidence level (CL); this is the first such limit. The limit
is calculated assuming a Gaussian distribution as the central value plus 1.28
times the addition in quadrature of the statistical and systematic
uncertainties. This branching ratio is predicted to be in the range
$(1-3)\times 10^{-6}$ if the $f_{0}(980)$ resonance is formed of tetra-quarks,
but can be much smaller if the $f_{0}(980)$ is a standard quark anti-quark
resonance [19]. Our limit is at the lower boundary of the tetra-quark
prediction, and is consistent with a quark anti-quark resonance with a small
mixing angle. In Section 7.2, we show that the mixing angle, describing the
admixture of $s\bar{s}$ and light quarks, is less than 31∘ at 90% CL.

The other branching fractions are consistent with and more precise than the
previous measurements from BaBar [26, *Aubert:2007xw]. Using ${\cal
B}(\omega\rightarrow\pi^{+}\pi^{-})=(1.53^{+0.11}_{-0.13})\%$ [31], we measure

$\frac{{\cal B}(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\omega)}{{\cal B}(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\rho^{0})}=0.61^{+0.24+0.31}_{-0.14-0.16},$

and

${\cal B}(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\omega)=(1.5_{-0.3-0.4}^{+0.6+0.7})\times 10^{-5}.$

This is consistent with the LHCb measurement $\frac{{\cal B}(\kern
1.25995pt\overline{\kern-1.25995ptB}{}^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\omega)}{{\cal B}(\kern
1.25995pt\overline{\kern-1.25995ptB}{}^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\rho^{0})}=0.89\pm 0.19^{+0.07}_{-0.13}$, using the
$\omega\rightarrow\pi^{+}\pi^{-}\pi^{0}$ mode [48].

Table 10: Branching fractions for each channel. The upper limit at 90% CL is also quoted for the $f_{0}(980)$ resonance which has a significance smaller than 3$\sigma$. The first uncertainty is statistical and the second the total systematic. Channel | ${\cal B}(\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}R,R\rightarrow\pi^{+}\pi^{-})$ | Upper limit of $\cal{B}$
---|---|---
|  | (at 90% CL)
$\rho(770)$ | $(2.49_{-0.13-0.23}^{+0.20+0.16})\times 10^{-5}$ | -
$\omega(782)$ | $(2.3_{-0.5-0.6}^{+0.9+1.1})\times 10^{-7}$ | -
$f_{0}(980)$ | $(6.1^{+3.1+1.7}_{-2.0-1.4})\times 10^{-7}$ | $<1.1\times 10^{-6}$
$f_{2}(1270)$ | $(3.5\pm 0.4\pm 0.4)\times 10^{-6}$ | -
$\rho(1450)$ | $(2.1^{+1.0+2.2}_{-0.6-0.4})\times 10^{-6}$ | -
$f_{0}(500)$ | $(6.4\pm 0.8^{+2.4}_{-0.8})\times 10^{-6}$ | -

##  6 Systematic uncertainties

Table 11: Relative systematic uncertainties on branching fractions (%). Source | Uncertainty (%)
---|---
Tracking efficiency | 1.0
Material and physical effects | 2.0
Particle identification efficiency | 1.0
$\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ $p$ and $p_{\rm T}$ distributions | 0.5
$B^{-}$ $p$ and $p_{\rm T}$ distributions | 0.5
Dalitz modeling | 0.6
Background modeling | 0.5
Sum of above sources | 2.7
$\mathcal{B}(B^{-}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{-})$ | 4.1
Total | 4.9

The contributions to the systematic uncertainties on the branching fractions
are listed in Table 11. Since the branching fractions are measured with
respect to the $B^{-}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{-}$ mode, which has a different number of charged tracks than the
decays of interest, a 1% systematic uncertainty is assigned due to differences
in the tracking performance between data and simulation. Another 2%
uncertainty is assigned because of the difference between two pions and one
kaon in the final states, due to decay in flight, multiple scattering, and
hadronic interactions. Small uncertainties are introduced if the simulation
does not have the correct $B$ meson kinematic distributions. We are relatively
insensitive to any differences in the $B$ meson $p$ and $p_{\rm T}$
distributions since we are measuring the relative rates. By varying the $p$
and $p_{\rm T}$ distributions we see at most a change of 0.5%. There is a 1.0%
systematic uncertainty assigned for the relative particle identification
efficiencies (0.5% per particle). These efficiencies have been corrected from
those predicted in the simulation by using the data from
$D^{*+}\rightarrow\pi^{+}D^{0}(\rightarrow K^{-}\pi^{+})$. A 0.6% uncertainty
is included for the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\pi^{-}\pi^{+}$ efficiency, estimated by changing the best model to that
including all possible resonances. The $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ signal yield is changed by 0.5%
when the shape of the combinatorial background is changed from an exponential
to a linear function. The total systematic uncertainty is obtained by adding
each source of systematic uncertainty in quadrature as they are uncorrelated.
In addition, the largest source is $4.1\%$ due to the uncertainty of
$\mathcal{B}(B^{-}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{-})$ which is quoted separately.

Table 12: Absolute systematic uncertainties on the results of the Dalitz analysis. Item | Acceptance | Background | Fit model | Resonance parameters | Total
---|---|---|---|---|---
Fit fractions (%)
$\rho(770)$ | $\pm 0.9$ | ${}^{+2.0}_{-3.1}$ | ${}^{+0.6}_{-3.2}$ | $\pm 1.6$ | ${}^{+2.8}_{-4.8}$
$\omega(782)$ | $\pm 0.08$ | ${}^{+0.23}_{-0.06}$ | ${}^{+0.11}_{-0.10}$ | ${}^{+0.028}_{-0.014}$ | ${}_{-0.14}^{+0.27}$
$f_{0}(980)$ | $\pm 0.03$ | ${}_{-0.17}^{+0.24}$ | ${}^{+0.21}_{-0.18}$ | ${}_{-0.24}^{+0.29}$ | ${}^{+0.43}_{-0.35}$
$f_{2}(1270)$ | $\pm 0.06$ | ${}_{-0.59}^{+0.45}$ | ${}^{+0.85}_{-0.76}$ | $\pm 0.36$ | $\pm 1.0$
$\rho(1450)$ | $\pm 0.10$ | ${}^{+0.5}_{-0.6}$ | ${}^{+5.6}_{-0.7}$ | ${}^{+0.4}_{-0.3}$ | ${}^{+5.6}_{-0.9}$
$f_{0}(500)$ | $\pm 0.4$ | ${}_{-0.9}^{+1.6}$ | ${}^{+5.7}_{-1.6}$ | $\pm 0.6$ | ${}^{+6.0}_{-2.0}$
$\lambda=0$ fractions (%)
$\rho(770)$ | $\pm 1.0$ | ${}_{-2.0}^{+1.7}$ | ${}^{+4.9}_{-1.5}$ | $\pm 2.1$ | ${}^{+5.7}_{-3.4}$
$\omega(782)$ | $\pm 1.5$ | ${}_{-1.8}^{+3.5}$ | ${}^{+2.8}_{-3.5}$ | ${}_{-1.7}^{+1.2}$ | ${}^{+4.9}_{-4.5}$
$f_{2}(1270)$ | $\pm 0.3$ | $\pm 2.4$ | ${}^{+3.7}_{-3.4}$ | $\pm 1.5$ | $\pm 4.5$
$\rho(1450)$ | $\pm 0.9$ | ${}_{-8.4}^{+4.8}$ | ${}^{+5.5}_{-5.1}$ | ${}^{+4.2}_{-6.1}$ | ${}^{+8.4}_{-11.6}$
Ratio of fit fractions (%)
$\omega(782)/\rho(770)$ | $\pm 0.13$ | ${}_{-0.11}^{+0.41}$ | ${}_{-0.16}^{+0.18}$ | ${}_{-0.022}^{+0.034}$ | ${}^{+0.47}_{-0.23}$
$f_{0}(980)/f_{0}(500)$ | $\pm 0.3$ | ${}^{+1.5}_{-1.1}$ | ${}^{+1.0}_{-2.1}$ | ${}_{-1.8}^{+2.2}$ | $\pm 3.0$

The sources of the systematic uncertainties on the results of the Dalitz plot
analysis are summarized in Table 12. For the uncertainties due to the
acceptance or background modeling, we repeat the data fit 100 times where the
parameters of acceptance or background modeling are generated according to the
corresponding covariance matrix. We also study the acceptance function by
changing the minimum IP $\chi^{2}$ requirement from 9 to 12.5 on both of the
pion candidates. As shown previously [10], this increases the $\chi^{2}$ of
the fit to the angular distributions by one unit. The acceptance function is
then applied to the data with the original minimum IP $\chi^{2}$ selection of
9, and the likelihood fit is redone and the uncertainties are estimated by
comparing the results with the best fit model. The larger of the two
variations is taken as uncertainty due to the acceptance.

We study the effect of ignoring the experimental mass resolution in the fit by
comparing fits between different pseudo-experiments with and without the
resolution included. As the widths of the resonances we consider are much
larger than the mass resolution, we find that the effects are negligible
except for the $\omega(782)$ resonance whose fit fraction is underestimated by
($0.09\pm 0.08$)%. Thus, we apply a $0.09\%$ correction to the $\omega(782)$
fraction and assign an additional $\pm 0.08\%$ in the acceptance systematic
uncertainty. The results shown in the previous sections already include this
correction.

In the default fit, the signal fraction $f_{\rm sig}=0.621\pm 0.009$, defined
in Eq. 1 is fixed; we vary its value within its error to estimate the
systematic uncertainty. The change is added in quadrature with the background
modeling uncertainties.

The uncertainties due to the fit model include adding each resonance that is
listed in Table 4 but not used in the best model, changing the default values
of $L_{B}$ in P- and D-wave cases, varying the hadron scale $r$ parameters for
the $B$ meson and $R$ resonance to $3.0\mathrm{\,Ge\kern-1.00006ptV}^{-1}$ for
both, replacing the $f_{0}(500)$ model by a Zhou and Bugg function [49, 50]
and using the alternate Gounaris and Sakurai model [51] for $\rho$ resonances.
Then the largest variations among those changes are assigned as the systematic
uncertainties for modeling (see Table 12).

Finally, we repeat the data fit by varying the mass and width of resonances
(see Table 4) within their errors one at a time, and add the changes in
quadrature.

##  7 Further results and implications

###  7.1 Resonant structure

The largest intermediate state in $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\pi^{+}\pi^{-}$ decays is the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\rho(770)$ mode. Beside the $\rho(770)$, significant $f_{2}(1270)$ and
$f_{0}(500)$ contributions are also seen. The smaller $\omega(782)$ and
$\rho(1450)$ resonances have 3.1$\sigma$ and 3.2$\sigma$ significances
respectively, including systematic uncertainties. The systematic uncertainties
reduce the significance of the $f_{0}(980)$ to below $3\sigma$. Replacing the
$f_{0}(500)$ by a non-resonant component increases $-\ln{\cal L}$ by 117, and
worsens the $\chi^{2}$ by 192 with the same ndf resulting in a fit confidence
level of $1.8\times 10^{-7}$. Thus the $f_{0}(500)$ state is firmly
established in $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\pi^{+}\pi^{-}$ decays.

As discussed in the introduction, a region with only S- and P-waves is
preferred for measuring $\sin 2\beta^{\rm eff}$. The best fit model
demonstrates that the mass region within $\pm
149$$\mathrm{\,Me\kern-1.00006ptV}$ (one full width) of the $\rho(770)$ mass
contains only $(0.72\pm 0.09)\%$ D-wave contribution, thus this region can be
used for a clean $C\\!P$ measurement. The S-wave in this region is
(11.9$\pm$1.7)%, where the fraction is the sum of individual fit fractions and
the interference.

###  7.2 Mixing angle between $f_{0}(980)$ and $f_{0}(500)$

The scalar nonet is quite an enigma. The mysteries are summarized in Ref.
[52], and in the “Note on scalar mesons” in the PDG [31]. Let us contrast the
masses of the lightest vector mesons with those of the scalars, listed in
Table 13.

Table 13: Masses of light vector and scalar resonances. All values are taken from [31], except for the $f_{0}(500)$ [43]. Isospin | Vector particle | Vector mass ($\mathrm{\,Me\kern-1.00006ptV}$) | Scalar particle | Scalar mass ($\mathrm{\,Me\kern-1.00006ptV}$)
---|---|---|---|---
0 | $\omega$ | 783 | $f_{0}(500)$ | 513
1 | $\rho$ | 776 | $a_{0}$ | 980
1/2 | $K^{*}$ | 980 | $\kappa$ | 800
0 | $\phi$ | 1020 | $f_{0}$ | 980

For the vector particles, the $\omega$ and $\rho$ masses are nearly degenerate
and the masses increase as the $s$-quark content increases. For the scalar
particles, however, the mass dependence differs in several ways which requires
an explanation. Some authors introduce the concept of $q\bar{q}q\bar{q}$
states or superpositions of the four-quark state with the $q\bar{q}$ state. In
either case, the $I=0$ $f_{0}(500)$ and the $f_{0}(980)$ are thought to be
mixtures of the underlying states whose mixing angle has been estimated
previously (see Ref. [19] and references contained therein).

The mixing is parameterized by a 2$\times$2 rotation matrix characterized by
the angle $\varphi_{m}$, giving in our case

$\displaystyle|f_{0}(980)\rangle$ $\displaystyle=$
$\displaystyle\;\;\;\cos\varphi_{m}|s\bar{s}\rangle+\sin\varphi_{m}|n\bar{n}\rangle$
$\displaystyle|f_{0}(500)\rangle$ $\displaystyle=$
$\displaystyle-\sin\varphi_{m}|s\bar{s}\rangle+\cos\varphi_{m}|n\bar{n}\rangle,$
$\displaystyle{\rm where~{}}|n\bar{n}\rangle$ $\displaystyle\equiv$
$\displaystyle\frac{1}{\sqrt{2}}\left(|u\bar{u}\rangle+|d\bar{d}\rangle\right).$
(32)

In this case only the $|d\bar{d}\rangle$ part of the $|n\bar{n}\rangle$ wave
function contributes (see Fig. 2). Thus we have

$\tan^{2}\varphi_{m}=\frac{{\cal{B}}\left(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}f_{0}(980)\right)}{{\cal{B}}\left(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}f_{0}(500)\right)}\frac{\Phi(500)}{\Phi(980)},$ (33)

where the $\Phi$ terms denote the phase space factors. The phase space in this
pseudoscalar to vector-pseudoscalar decay is proportional to the cube of the
$f_{0}$ three-momentum. Taking the average of the momentum dependent phase
space over the resonant line shapes results in the ratio of phase space
factors $\frac{\Phi(500)}{\Phi(980)}$ being equal to 1.25.

Using the data shown in Table 10 we determine the ratio of branching fractions
for both resonances resulting in the $\pi^{+}\pi^{-}$ final state as

$\frac{{\cal{B}}\left(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}f_{0}(980)\right)\times{\cal{B}}\left(f_{0}(980)\rightarrow\pi^{+}\pi^{-}\right)}{{\cal{B}}\left(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}f_{0}(500)\right)\times{\cal{B}}\left(f_{0}(500)\rightarrow\pi^{+}\pi^{-}\right)}=(9.5^{+6.7}_{-3.4}\pm
3.0)\%.$

This value must be corrected for the individual branching fractions of the
$f_{0}$ resonances into the $\pi^{+}\pi^{-}$ final state.

BaBar has measured $\frac{{\cal{B}}\left(f_{0}(980)\rightarrow
K^{+}K^{-}\right)}{{\cal{B}}\left(f_{0}(980)\rightarrow\pi^{+}\pi^{-}\right)}=0.69\pm
0.32$ using $B\rightarrow KKK$ and $B\rightarrow K\pi\pi$ decays [53]. BES
obtained relative branching ratios using $\psi(2S)\rightarrow\gamma\chi_{c0}$
decays where the $\chi_{c0}\rightarrow f_{0}(980)f_{0}(980)$, and either both
$f_{0}(980)$ candidates decay into $\pi^{+}\pi^{-}$ or one into
$\pi^{+}\pi^{-}$ and the other into $K^{+}K^{-}$ pairs [54, *Ablikim:2005kp].
From their results we obtain $\frac{{\cal{B}}\left(f_{0}(980)\rightarrow
K^{+}K^{-}\right)}{{\cal{B}}\left(f_{0}(980)\rightarrow\pi^{+}\pi^{-}\right)}=0.25^{+0.17}_{-0.11}$[56].
Averaging the two measurements gives

$\frac{{\cal{B}}\left(f_{0}(980)\rightarrow
K^{+}K^{-}\right)}{{\cal{B}}\left(f_{0}(980)\rightarrow\pi^{+}\pi^{-}\right)}=0.35_{-0.14}^{+0.15}\,.$
(34)

Assuming that the $\pi\pi$ and $KK$ decays are dominant we obtain

${\cal{B}}\left(f_{0}(980)\rightarrow\pi^{+}\pi^{-}\right)=\left(46\pm
6\right)\%,$ (35)

where we have assumed that the only other decays are to $\pi^{0}\pi^{0}$, half
of the $\pi^{+}\pi^{-}$ rate, and to neutral kaons, taken equal to charged
kaons. We use
${\cal{B}}\left(f_{0}(500)\rightarrow\pi^{+}\pi^{-}\right)=\frac{2}{3}$, which
results from isospin Clebsch-Gordon coefficients, and assuming that the only
decays are into two pions. Since we have only an upper limit on the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}f_{0}(980)$ final state, we will
only find an upper limit on the mixing angle, so if any other decay modes of
the $f_{0}(500)$ ($f_{0}(980)$) exist, they would make the limit more (less)
stringent. Our limit then is

$\tan^{2}\varphi_{m}=\frac{{\cal{B}}\left(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}f_{0}(980)\right)}{{\cal{B}}\left(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}f_{0}(500)\right)}\frac{\Phi(500)}{\Phi(980)}<0.35~{}{\rm
at~{}90\%~{}confidence~{}level},$

which translates into a limit

$|\varphi_{m}|<31^{\circ}~{}{\rm at~{}90\%~{}confidence~{}level}.$

Various mixing angle measurements have been derived in the literature and
summarized in Ref. [19]. There are a wide range of values including: (a) using
$D_{s}^{+}\rightarrow\pi^{+}\pi^{+}\pi^{-}$ transitions which give a range
$35^{\circ}<|\varphi_{m}|<55^{\circ}$, (b) using radiative decays where two
solutions were found either $\varphi_{m}=4^{\circ}\pm 3^{\circ}$ or
$136^{\circ}\pm 6^{\circ}$, (c) using resonance decays from both
$\phi\rightarrow\gamma\pi^{0}\pi^{0}$ and
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\rightarrow\omega\pi\pi$ where a
value of $\varphi_{m}\simeq 20^{\circ}$ was found, (d) using the $D^{\pm}$ and
$D_{s}^{\pm}$ decays into $f_{0}(980)\pi^{\pm}$ and $f_{0}(980)K^{\pm}$ where
values of $\varphi_{m}=31^{\circ}\pm 5^{\circ}$ or $42^{\circ}\pm 7^{\circ}$
were found.

##  8 Conclusions

We have studied the resonance structure of $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\pi^{+}\pi^{-}$ using a modified Dalitz plot analysis where we also
include the decay angle of the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$
meson. The decay distributions are formed from a series of final states
described by individual $\pi^{+}\pi^{-}$ interfering decay amplitudes. The
largest component is the $\rho(770)$ resonance. The data are best described by
adding the $f_{2}(1270)$, $f_{0}(500)$, $\omega(782)$, $\rho(1450)$ and
$f_{0}(980)$ resonances, where the $f_{0}(980)$ resonance contributes less
than $3\sigma$ significance. The results are listed in Table 6.

We set an upper limit ${\cal{B}}\left(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}f_{0}(980)\right)\times{\cal{B}}\left(f_{0}(980)\rightarrow\pi^{+}\pi^{-}\right)<1.1\times
10^{-6}$ at 90% confidence level that favors somewhat a quark anti-quark
interpretation of the $f_{0}(980)$ resonance. We also have firmly established
the existence of the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}f_{0}(500)$
intermediate resonant state in $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ decays, and limit the absolute
value of the mixing angle between the two lightest scalar states to be less
than $31^{\circ}$ at 90% confidence level.

Our six-resonance best fit shows that the mass region within one full width of
the $\rho(770)$ contains mostly P-wave, $(11.9\pm 1.7)$% S-wave, and only
$(0.72\pm 0.09)$% D-wave. Thus this region can be used to perform $C\\!P$
violation measurements, as the S- and P-wave components can be treated in the
same manner as in the analysis of $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\phi$ [4, 5, 6, *Abazov:2011ry, *:2012fu]. The measured value of the
asymmetry can be compared to that found in other modes such as $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\kern 1.99997pt\overline{\kern-1.99997ptK}{}^{0}$ in order to ascertain
the possible effects due to penguin amplitudes.

The measured branching ratio is

${\cal B}(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\pi^{+}\pi^{-})=(3.97\pm 0.09\pm 0.11\pm 0.16)\times 10^{-5},$

where the first uncertainty is statistical, the second is systematic and the
third is due to the uncertainty of ${\cal
B}(B^{-}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{-})$. The
largest contribution is the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\rho(770)$ mode with a branching fraction of
$(2.49_{-0.13-0.23}^{+0.20+0.16})\times 10^{-5}$.

## Acknowledgements

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); NSFC
(China); CNRS/IN2P3 and Region Auvergne (France); BMBF, DFG, HGF and MPG
(Germany); SFI (Ireland); INFN (Italy); FOM and NWO (The Netherlands); SCSR
(Poland); ANCS/IFA (Romania); MinES, Rosatom, RFBR and NRC “Kurchatov
Institute” (Russia); MinECo, XuntaGal and GENCAT (Spain); SNSF and SER
(Switzerland); NAS Ukraine (Ukraine); STFC (United Kingdom); NSF (USA). We
also acknowledge the support received from the ERC under FP7. 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 thankful for the computing resources put at our disposal by
Yandex LLC (Russia), as well as to the communities behind the multiple open
source software packages that we depend on.

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