EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)

 CERN-PH-EP-2013-074 LHCb-PAPER-2013-019 8 July 2013

Differential branching fraction

and angular analysis of

the decay $B^{0}\\!\rightarrow K^{*0}\mu^{+}\mu^{-}$

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

The angular distribution and differential branching fraction of the decay
$B^{0}\\!\rightarrow K^{*0}\mu^{+}\mu^{-}$ are studied using a data sample,
collected by the LHCb experiment in $pp$ collisions at
$\sqrt{s}=7\mathrm{\,Te\kern-1.00006ptV}$, corresponding to an integrated
luminosity of $1.0\mbox{\,fb}^{-1}$. Several angular observables are measured
in bins of the dimuon invariant mass squared, $q^{2}$. A first measurement of
the zero-crossing point of the forward-backward asymmetry of the dimuon system
is also presented. The zero-crossing point is measured to be $q_{0}^{2}=4.9\pm
0.9\mathrm{\,Ge\kern-1.00006ptV}^{2}/c^{4}$, where the uncertainty is the sum
of statistical and systematic uncertainties. The results are consistent with
the Standard Model predictions.

Submitted to JHEP

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

LHCb collaboration

R. Aaij40, C. Abellan Beteta35,n, B. Adeva36, M. Adinolfi45, C. Adrover6, A.
Affolder51, Z. Ajaltouni5, J. Albrecht9, F. Alessio37, M. Alexander50, S.
Ali40, G. Alkhazov29, P. Alvarez Cartelle36, A.A. Alves Jr24,37, S. Amato2, S.
Amerio21, Y. Amhis7, L. Anderlini17,f, J. Anderson39, R. Andreassen56, R.B.
Appleby53, O. Aquines Gutierrez10, F. Archilli18, A. Artamonov 34, M.
Artuso58, E. Aslanides6, G. Auriemma24,m, S. Bachmann11, J.J. Back47, C.
Baesso59, V. Balagura30, W. Baldini16, R.J. Barlow53, C. Barschel37, S.
Barsuk7, W. Barter46, Th. Bauer40, A. Bay38, J. Beddow50, F. Bedeschi22, I.
Bediaga1, S. Belogurov30, K. Belous34, I. Belyaev30, E. Ben-Haim8, G.
Bencivenni18, S. Benson49, J. Benton45, A. Berezhnoy31, R. Bernet39, M.-O.
Bettler46, M. van Beuzekom40, A. Bien11, S. Bifani44, T. Bird53, A.
Bizzeti17,h, P.M. Bjørnstad53, T. Blake37, F. Blanc38, J. Blouw11, S. Blusk58,
V. Bocci24, A. Bondar33, N. Bondar29, W. Bonivento15, S. Borghi53, A.
Borgia58, T.J.V. Bowcock51, E. Bowen39, C. Bozzi16, T. Brambach9, J. van den
Brand41, J. Bressieux38, D. Brett53, M. Britsch10, T. Britton58, N.H. Brook45,
H. Brown51, I. Burducea28, A. Bursche39, G. Busetto21,q, J. Buytaert37, S.
Cadeddu15, O. Callot7, M. Calvi20,j, M. Calvo Gomez35,n, A. Camboni35, P.
Campana18,37, D. Campora Perez37, A. Carbone14,c, G. Carboni23,k, R.
Cardinale19,i, A. Cardini15, H. Carranza-Mejia49, L. Carson52, K. Carvalho
Akiba2, G. Casse51, L. Castillo Garcia37, M. Cattaneo37, Ch. Cauet9, M.
Charles54, Ph. Charpentier37, P. Chen3,38, N. Chiapolini39, M. Chrzaszcz 25,
K. Ciba37, X. Cid Vidal37, G. Ciezarek52, P.E.L. Clarke49, M. Clemencic37,
H.V. Cliff46, J. Closier37, C. Coca28, V. Coco40, J. Cogan6, E. Cogneras5, P.
Collins37, A. Comerma-Montells35, A. Contu15,37, A. Cook45, M. Coombes45, S.
Coquereau8, G. Corti37, B. Couturier37, G.A. Cowan49, D.C. Craik47, S.
Cunliffe52, R. Currie49, C. D’Ambrosio37, P. David8, P.N.Y. David40, A.
Davis56, I. De Bonis4, K. De Bruyn40, S. De Capua53, M. De Cian39, J.M. De
Miranda1, L. De Paula2, W. De Silva56, P. De Simone18, D. Decamp4, M.
Deckenhoff9, L. Del Buono8, N. Déléage4, D. Derkach14, O. Deschamps5, F.
Dettori41, A. Di Canto11, F. Di Ruscio23,k, H. Dijkstra37, M. Dogaru28, S.
Donleavy51, F. Dordei11, A. Dosil Suárez36, D. Dossett47, A. Dovbnya42, F.
Dupertuis38, R. Dzhelyadin34, A. Dziurda25, A. Dzyuba29, S. Easo48,37, U.
Egede52, V. Egorychev30, S. Eidelman33, D. van Eijk40, S. Eisenhardt49, U.
Eitschberger9, R. Ekelhof9, L. Eklund50,37, I. El Rifai5, Ch. Elsasser39, D.
Elsby44, A. Falabella14,e, C. Färber11, G. Fardell49, C. Farinelli40, S.
Farry12, V. Fave38, D. Ferguson49, V. Fernandez Albor36, F. Ferreira
Rodrigues1, M. Ferro-Luzzi37, S. Filippov32, M. Fiore16, C. Fitzpatrick37, M.
Fontana10, F. Fontanelli19,i, R. Forty37, O. Francisco2, M. Frank37, C.
Frei37, M. Frosini17,f, S. Furcas20, E. Furfaro23,k, A. Gallas Torreira36, D.
Galli14,c, M. Gandelman2, P. Gandini58, Y. Gao3, J. Garofoli58, P. Garosi53,
J. Garra Tico46, L. Garrido35, C. Gaspar37, R. Gauld54, E. Gersabeck11, M.
Gersabeck53, T. Gershon47,37, Ph. Ghez4, V. Gibson46, V.V. Gligorov37, C.
Göbel59, D. Golubkov30, A. Golutvin52,30,37, A. Gomes2, H. Gordon54, M.
Grabalosa Gándara5, R. Graciani Diaz35, L.A. Granado Cardoso37, E. Graugés35,
G. Graziani17, A. Grecu28, E. Greening54, S. Gregson46, P. Griffith44, O.
Grünberg60, B. Gui58, E. Gushchin32, Yu. Guz34,37, T. Gys37, C.
Hadjivasiliou58, G. Haefeli38, C. Haen37, S.C. Haines46, S. Hall52, T.
Hampson45, S. Hansmann-Menzemer11, N. Harnew54, S.T. Harnew45, J. Harrison53,
T. Hartmann60, J. He37, V. Heijne40, K. Hennessy51, P. Henrard5, J.A. Hernando
Morata36, E. van Herwijnen37, E. Hicks51, D. Hill54, M. Hoballah5, C.
Hombach53, P. Hopchev4, W. Hulsbergen40, P. Hunt54, T. Huse51, N. Hussain54,
D. Hutchcroft51, D. Hynds50, V. Iakovenko43, M. Idzik26, P. Ilten12, R.
Jacobsson37, A. Jaeger11, E. Jans40, P. Jaton38, A. Jawahery57, F. Jing3, M.
John54, D. Johnson54, C.R. Jones46, C. Joram37, B. Jost37, M. Kaballo9, S.
Kandybei42, M. Karacson37, T.M. Karbach37, I.R. Kenyon44, U. Kerzel37, T.
Ketel41, A. Keune38, B. Khanji20, O. Kochebina7, I. Komarov38, R.F. Koopman41,
P. Koppenburg40, M. Korolev31, A. Kozlinskiy40, L. Kravchuk32, K. Kreplin11,
M. Kreps47, G. Krocker11, P. Krokovny33, F. Kruse9, M. Kucharczyk20,25,j, V.
Kudryavtsev33, T. Kvaratskheliya30,37, V.N. La Thi38, D. Lacarrere37, G.
Lafferty53, A. Lai15, D. Lambert49, R.W. Lambert41, E. Lanciotti37, G.
Lanfranchi18, C. Langenbruch37, T. Latham47, C. Lazzeroni44, R. Le Gac6, J.
van Leerdam40, J.-P. Lees4, R. Lefèvre5, A. Leflat31, J. Lefrançois7, S.
Leo22, O. Leroy6, T. Lesiak25, B. Leverington11, Y. Li3, L. Li Gioi5, M.
Liles51, R. Lindner37, C. Linn11, B. Liu3, G. Liu37, S. Lohn37, I.
Longstaff50, J.H. Lopes2, E. Lopez Asamar35, N. Lopez-March38, H. Lu3, D.
Lucchesi21,q, J. Luisier38, H. Luo49, F. Machefert7, I.V. Machikhiliyan4,30,
F. Maciuc28, O. Maev29,37, S. Malde54, G. Manca15,d, G. Mancinelli6, U.
Marconi14, R. Märki38, J. Marks11, G. Martellotti24, A. Martens8, L. Martin54,
A. Martín Sánchez7, M. Martinelli40, D. Martinez Santos41, D. Martins Tostes2,
A. Massafferri1, R. Matev37, Z. Mathe37, C. Matteuzzi20, E. Maurice6, A.
Mazurov16,32,37,e, J. McCarthy44, A. McNab53, R. McNulty12, B. Meadows56,54,
F. Meier9, M. Meissner11, M. Merk40, D.A. Milanes8, M.-N. Minard4, J. Molina
Rodriguez59, S. Monteil5, D. Moran53, P. Morawski25, M.J. Morello22,s, R.
Mountain58, I. Mous40, F. Muheim49, K. Müller39, R. Muresan28, B. Muryn26, B.
Muster38, P. Naik45, T. Nakada38, R. Nandakumar48, I. Nasteva1, M. Needham49,
N. Neufeld37, A.D. Nguyen38, T.D. Nguyen38, C. Nguyen-Mau38,p, M. Nicol7, V.
Niess5, R. Niet9, N. Nikitin31, T. Nikodem11, A. Nomerotski54, A. Novoselov34,
A. Oblakowska-Mucha26, V. Obraztsov34, S. Oggero40, S. Ogilvy50, O.
Okhrimenko43, R. Oldeman15,d, M. Orlandea28, J.M. Otalora Goicochea2, P.
Owen52, A. Oyanguren 35,o, B.K. Pal58, A. Palano13,b, M. Palutan18, J.
Panman37, A. Papanestis48, M. Pappagallo50, C. Parkes53, C.J. Parkinson52, G.
Passaleva17, G.D. Patel51, M. Patel52, G.N. Patrick48, C. Patrignani19,i, C.
Pavel-Nicorescu28, A. Pazos Alvarez36, A. Pellegrino40, G. Penso24,l, M. Pepe
Altarelli37, S. Perazzini14,c, D.L. Perego20,j, E. Perez Trigo36, A. Pérez-
Calero Yzquierdo35, P. Perret5, M. Perrin-Terrin6, G. Pessina20, K.
Petridis52, A. Petrolini19,i, A. Phan58, E. Picatoste Olloqui35, B. Pietrzyk4,
T. Pilař47, D. Pinci24, S. Playfer49, M. Plo Casasus36, F. Polci8, G. Polok25,
A. Poluektov47,33, E. Polycarpo2, A. Popov34, D. Popov10, B. Popovici28, C.
Potterat35, A. Powell54, J. Prisciandaro38, V. Pugatch43, A. Puig Navarro38,
G. Punzi22,r, W. Qian4, J.H. Rademacker45, B. Rakotomiaramanana38, M.S.
Rangel2, I. Raniuk42, N. Rauschmayr37, G. Raven41, S. Redford54, M.M. Reid47,
A.C. dos Reis1, S. Ricciardi48, A. Richards52, K. Rinnert51, V. Rives
Molina35, D.A. Roa Romero5, P. Robbe7, E. Rodrigues53, P. Rodriguez Perez36,
S. Roiser37, V. Romanovsky34, A. Romero Vidal36, J. Rouvinet38, T. Ruf37, F.
Ruffini22, H. Ruiz35, P. Ruiz Valls35,o, G. Sabatino24,k, J.J. Saborido
Silva36, N. Sagidova29, P. Sail50, B. Saitta15,d, V. Salustino Guimaraes2, C.
Salzmann39, B. Sanmartin Sedes36, M. Sannino19,i, R. Santacesaria24, C.
Santamarina Rios36, E. Santovetti23,k, M. Sapunov6, A. Sarti18,l, C.
Satriano24,m, A. Satta23, M. Savrie16,e, D. Savrina30,31, P. Schaack52, M.
Schiller41, H. Schindler37, M. Schlupp9, M. Schmelling10, B. Schmidt37, O.
Schneider38, A. Schopper37, M.-H. Schune7, R. Schwemmer37, B. Sciascia18, A.
Sciubba24, M. Seco36, A. Semennikov30, K. Senderowska26, I. Sepp52, N.
Serra39, J. Serrano6, P. Seyfert11, M. Shapkin34, I. Shapoval16,42, P.
Shatalov30, Y. Shcheglov29, T. Shears51,37, L. Shekhtman33, O. Shevchenko42,
V. Shevchenko30, A. Shires52, R. Silva Coutinho47, T. Skwarnicki58, N.A.
Smith51, E. Smith54,48, M. Smith53, M.D. Sokoloff56, F.J.P. Soler50, F.
Soomro18, D. Souza45, B. Souza De Paula2, B. Spaan9, A. Sparkes49, P.
Spradlin50, F. Stagni37, S. Stahl11, O. Steinkamp39, S. Stoica28, S. Stone58,
B. Storaci39, M. Straticiuc28, U. Straumann39, V.K. Subbiah37, L. Sun56, S.
Swientek9, V. Syropoulos41, M. Szczekowski27, P. Szczypka38,37, T. Szumlak26,
S. T’Jampens4, M. Teklishyn7, E. Teodorescu28, F. Teubert37, C. Thomas54, E.
Thomas37, J. van Tilburg11, V. Tisserand4, M. Tobin38, S. Tolk41, D.
Tonelli37, S. Topp-Joergensen54, N. Torr54, E. Tournefier4,52, S. Tourneur38,
M.T. Tran38, M. Tresch39, A. Tsaregorodtsev6, P. Tsopelas40, N. Tuning40, M.
Ubeda Garcia37, A. Ukleja27, D. Urner53, U. Uwer11, V. Vagnoni14, G.
Valenti14, R. Vazquez Gomez35, P. Vazquez Regueiro36, S. Vecchi16, J.J.
Velthuis45, M. Veltri17,g, G. Veneziano38, M. Vesterinen37, B. Viaud7, D.
Vieira2, X. Vilasis-Cardona35,n, A. Vollhardt39, D. Volyanskyy10, D. Voong45,
A. Vorobyev29, V. Vorobyev33, C. Voß60, H. Voss10, R. Waldi60, R. Wallace12,
S. Wandernoth11, J. Wang58, D.R. Ward46, N.K. Watson44, A.D. Webber53, D.
Websdale52, M. Whitehead47, J. Wicht37, J. Wiechczynski25, D. Wiedner11, L.
Wiggers40, G. Wilkinson54, M.P. Williams47,48, M. Williams55, F.F. Wilson48,
J. Wishahi9, M. Witek25, S.A. Wotton46, S. Wright46, S. Wu3, K. Wyllie37, Y.
Xie49,37, F. Xing54, Z. Xing58, Z. Yang3, R. Young49, X. Yuan3, O.
Yushchenko34, M. Zangoli14, M. Zavertyaev10,a, F. Zhang3, L. Zhang58, W.C.
Zhang12, Y. Zhang3, A. Zhelezov11, A. Zhokhov30, L. Zhong3, A. Zvyagin37.

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 Padova, Padova, Italy
22Sezione INFN di Pisa, Pisa, Italy
23Sezione INFN di Roma Tor Vergata, Roma, Italy
24Sezione INFN di Roma La Sapienza, Roma, Italy
25Henryk Niewodniczanski Institute of Nuclear Physics Polish Academy of
Sciences, Kraków, Poland
26AGH - University of Science and Technology, Faculty of Physics and Applied
Computer Science, Kraków, Poland
27National Center for Nuclear Research (NCBJ), Warsaw, Poland
28Horia Hulubei National Institute of Physics and Nuclear Engineering,
Bucharest-Magurele, Romania
29Petersburg Nuclear Physics Institute (PNPI), Gatchina, Russia
30Institute of Theoretical and Experimental Physics (ITEP), Moscow, Russia
31Institute of Nuclear Physics, Moscow State University (SINP MSU), Moscow,
Russia
32Institute for Nuclear Research of the Russian Academy of Sciences (INR RAN),
Moscow, Russia
33Budker Institute of Nuclear Physics (SB RAS) and Novosibirsk State
University, Novosibirsk, Russia
34Institute for High Energy Physics (IHEP), Protvino, Russia
35Universitat de Barcelona, Barcelona, Spain
36Universidad de Santiago de Compostela, Santiago de Compostela, Spain
37European Organization for Nuclear Research (CERN), Geneva, Switzerland
38Ecole Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland
39Physik-Institut, Universität Zürich, Zürich, Switzerland
40Nikhef National Institute for Subatomic Physics, Amsterdam, The Netherlands
41Nikhef National Institute for Subatomic Physics and VU University Amsterdam,
Amsterdam, The Netherlands
42NSC Kharkiv Institute of Physics and Technology (NSC KIPT), Kharkiv, Ukraine
43Institute for Nuclear Research of the National Academy of Sciences (KINR),
Kyiv, Ukraine
44University of Birmingham, Birmingham, United Kingdom
45H.H. Wills Physics Laboratory, University of Bristol, Bristol, United
Kingdom
46Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom
47Department of Physics, University of Warwick, Coventry, United Kingdom
48STFC Rutherford Appleton Laboratory, Didcot, United Kingdom
49School of Physics and Astronomy, University of Edinburgh, Edinburgh, United
Kingdom
50School of Physics and Astronomy, University of Glasgow, Glasgow, United
Kingdom
51Oliver Lodge Laboratory, University of Liverpool, Liverpool, United Kingdom
52Imperial College London, London, United Kingdom
53School of Physics and Astronomy, University of Manchester, Manchester,
United Kingdom
54Department of Physics, University of Oxford, Oxford, United Kingdom
55Massachusetts Institute of Technology, Cambridge, MA, United States
56University of Cincinnati, Cincinnati, OH, United States
57University of Maryland, College Park, MD, United States
58Syracuse University, Syracuse, NY, United States
59Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), Rio de
Janeiro, Brazil, associated to 2
60Institut für Physik, Universität Rostock, Rostock, Germany, associated to 11

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
oIFIC, Universitat de Valencia-CSIC, Valencia, Spain
pHanoi University of Science, Hanoi, Viet Nam
qUniversità di Padova, Padova, Italy
rUniversità di Pisa, Pisa, Italy
sScuola Normale Superiore, Pisa, Italy

##  1 Introduction

The $B^{0}\\!\rightarrow K^{*0}\mu^{+}\mu^{-}$ decay,111Charge conjugation is
implied throughout this paper unless stated otherwise. where
$K^{*0}\\!\rightarrow K^{+}\pi^{-}$, is a $b\rightarrow s$ flavour changing
neutral current process that is mediated by electroweak box and penguin type
diagrams in the Standard Model (SM). The angular distribution of the
$K^{+}\pi^{-}\mu^{+}\mu^{-}$ system offers particular sensitivity to
contributions from new particles in extensions to the SM. The differential
branching fraction of the decay also provides information on the contribution
from those new particles but typically suffers from larger theoretical
uncertainties due to hadronic form factors.

The angular distribution of the decay can be described by three angles
($\theta_{\ell},\theta_{K}$ and $\phi$) and by the invariant mass squared of
the dimuon system ($q^{2}$). The $B^{0}\\!\rightarrow K^{*0}\mu^{+}\mu^{-}$
decay is self-tagging through the charge of the kaon and so there is some
freedom in the choice of the angular basis that is used to describe the decay.
In this paper, the angle $\theta_{\ell}$ is defined as the angle between the
direction of the $\mu^{+}$ ($\mu^{-}$) and the direction opposite that of the
$B^{0}$ ($\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}$) in the dimuon
rest frame. The angle $\theta_{K}$ is defined as the angle between the
direction of the kaon and the direction of opposite that of the $B^{0}$
($\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}$) in in the $K^{*0}$
($\kern 1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$) rest frame. The angle
$\phi$ is the angle between the plane containing the $\mu^{+}$ and $\mu^{-}$
and the plane containing the kaon and pion from the $K^{*0}$ ($\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$) in the $B^{0}$ ($\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}$) rest frame. The basis is designed
such that the angular definition for the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ decay is a $C\\!P$ transformation
of that for the $B^{0}$ decay. This basis differs from some that appear in the
literature. A graphical representation, and a more detailed description, of
the angular basis is given in Appendix A.

Using the notation of Ref. [1], the decay distribution of the $B^{0}$
corresponds to

$\begin{split}\frac{\mathrm{d}^{4}\Gamma}{\mathrm{d}q^{2}\,\mathrm{d}\cos\theta_{\ell}\,\mathrm{d}\cos\theta_{K}\,\mathrm{d}\phi}=\frac{9}{32\pi}&\left[\frac{}{}{I_{1}^{s}}\sin^{2}\theta_{K}+{I_{1}^{c}}\cos^{2}\theta_{K}~{}+\right.\\\
&\left.~{}\frac{}{}{I_{2}^{s}}\sin^{2}\theta_{K}\cos
2\theta_{\ell}+{I_{2}^{c}}\cos^{2}\theta_{K}\cos 2\theta_{\ell}~{}+\right.\\\
&\left.~{}\frac{}{}{I_{3}}\sin^{2}\theta_{K}\sin^{2}\theta_{\ell}\cos
2\phi+{{I_{4}\sin 2\theta_{K}\sin 2\theta_{\ell}\cos\phi}}~{}+\right.\\\
&~{}\frac{}{}\left.{{{I_{5}}\sin
2\theta_{K}\sin\theta_{\ell}\cos\phi}}+I_{6}\sin^{2}\theta_{K}\cos\theta_{\ell}~{}+\right.\\\
&~{}\frac{}{}\left.{{{I_{7}}\sin
2\theta_{K}\sin\theta_{\ell}\sin\phi}}+{{{I_{8}}\sin 2\theta_{K}\sin
2\theta_{\ell}\sin\phi}}~{}+\right.\\\
&~{}\frac{}{}\left.I_{9}\sin^{2}\theta_{K}\sin^{2}\theta_{\ell}\sin
2\phi\frac{}{}~{}\right]~{},\end{split}$ (1)

where the 11 coefficients, $I_{j}$, are bilinear combinations of $K^{*0}$
decay amplitudes, ${\cal A}_{m}$, and vary with $q^{2}$. The superscripts $s$
and $c$ in the first two terms arise in Ref. [1] and indicate either a
$\sin^{2}\theta_{K}$ or $\cos^{2}\theta_{K}$ dependence of the corresponding
angular term. In the SM, there are seven complex decay amplitudes,
corresponding to different polarisation states of the $K^{*0}$ and chiralities
of the dimuon system. In the angular coefficients, the decay amplitudes appear
in the combinations $|{\cal A}_{m}|^{2}$, ${\rm Re}({\cal A}_{m}{\cal
A}_{n}^{*})$ and ${\rm Im}({\cal A}_{m}{\cal A}_{n}^{*})$. Combining $B^{0}$
and $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ decays, and assuming
there are equal numbers of each, it is possible to build angular observables
that depend on the average of, or difference between, the distributions for
the $B^{0}$ and $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ decay,

$S_{j}=\left.\left(I_{j}+\bar{I}_{j}\right)\middle/\frac{\mathrm{d}\Gamma}{\mathrm{d}q^{2}}\right.~{}\text{or}~{}\left.A_{j}=\left(I_{j}-\bar{I}_{j}\right)\middle/\frac{\mathrm{d}\Gamma}{\mathrm{d}q^{2}}\right.~{}.$
(2)

These observables are referred to below as $C\\!P$ averages or $C\\!P$
asymmetries and are normalised with respect to the combined differential decay
rate, $\mathrm{d}\Gamma/\mathrm{d}q^{2}$, of $B^{0}$ and $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ decays. The observables $S_{7}$,
$S_{8}$ and $S_{9}$ depend on combinations ${\rm Im}({\cal A}_{m}{\cal
A}_{n}^{*})$ and are suppressed by the small size of the strong phase
difference between the decay amplitudes. They are consequently expected to be
close to zero across the full $q^{2}$ range not only in the SM but also in
most extensions. However, the corresponding $C\\!P$ asymmetries, $A_{7}$,
$A_{8}$ and $A_{9}$, are not suppressed by the strong phases involved [2] and
remain sensitive to the effects of new particles.

If the $B^{0}$ and $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ decays
are combined using the angular basis in Appendix A, the resulting angular
distribution is sensitive to only the $C\\!P$ averages of each of the angular
terms. Sensitivity to $A_{7}$, $A_{8}$ and $A_{9}$ is achieved by flipping the
sign of $\phi$ ($\phi\rightarrow-\phi$) for the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ decay. This procedure results in a
combined $B^{0}$ and $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}$
angular distribution that is sensitive to the $C\\!P$ averages $S_{1}-S_{6}$
and the $C\\!P$ asymmetries of $A_{7}$, $A_{8}$ and $A_{9}$.

In the limit that the dimuon mass is large compared to the mass of the muons,
$q^{2}\gg 4m_{\mu}^{2}$, the $C\\!P$ average of $I_{1}^{c}$, $I_{1}^{s}$,
$I_{2}^{c}$ and $I_{2}^{s}$ ($S_{1}^{c}$, $S_{1}^{s}$, $S_{2}^{c}$ and
$S_{2}^{s}$) are related to the fraction of longitudinal polarisation of the
$K^{*0}$ meson, $F_{\rm L}$ ($S_{1}^{c}=-S_{2}^{c}=F_{\rm L}$ and
$\frac{4}{3}S_{1}^{s}=4S_{2}^{s}=1-F_{\rm L}$). The angular term, $I_{6}$ in
Eq. 1, which has a $\sin^{2}\theta_{K}\cos\theta_{\ell}$ dependence, generates
a forward-backward asymmetry of the dimuon system, $A_{\rm FB}$ [3] ($A_{\rm
FB}=\frac{3}{4}S_{6}$). The term $S_{3}$ is related to the asymmetry between
the two sets of transverse $K^{*0}$ amplitudes, referred to in literature as
$A_{\rm T}^{2}$ [4], where $S_{3}=\frac{1}{2}\left(1-F_{\rm L}\right)A_{\rm
T}^{2}$.

In the SM, $A_{\rm FB}$ varies as a function of $q^{2}$ and is known to change
sign. The $q^{2}$ dependence arises from the interplay between the different
penguin and box diagrams that contribute to the decay. The position of the
zero-crossing point of $A_{\rm FB}$ is a precision test of the SM since, in
the limit of large $K^{*0}$ energy, its prediction is free from form-factor
uncertainties [3]. At large recoil, low values of $q^{2}$, penguin diagrams
involving a virtual photon dominate. In this $q^{2}$ region, $A_{\rm T}^{2}$
is sensitive to the polarisation of the virtual photon which, in the SM, is
predominately left-handed, due to the nature of the charged-current
interaction. In many possible extensions of the SM however, the photon can be
both left- or right-hand polarised, leading to large enhancements of $A_{\rm
T}^{2}$ [4].

The one-dimensional $\cos\theta_{\ell}$ and $\cos\theta_{K}$ distributions
have previously been studied by the LHCb [5], BaBar [6], Belle [7] and CDF [8]
experiments with much smaller data samples. The CDF experiment has also
previously studied the $\phi$ angle. Even with the larger dataset available in
this analysis, it is not yet possible to fit the data for all 11 angular
terms. Instead, rather than examining the one dimensional projections as has
been done in previous analyses, the angle $\phi$ is transformed such that

$\hat{\phi}=\begin{cases}\phi+\pi&\text{~{}if~{}}\phi<0\\\
\phi&\text{~{}otherwise}\end{cases}$ (3)

to cancel terms in Eq. 1 that have either a $\sin\phi$ or a $\cos\phi$
dependence. This provides a simplified angular expression, which contains only
$F_{\rm L}$, $A_{\rm FB}$, $S_{3}$ and $A_{9}$,

$\begin{split}\frac{1}{\mathrm{d}\Gamma/\mathrm{d}q^{2}}\frac{\mathrm{d}^{4}\Gamma}{\mathrm{d}q^{2}\,\mathrm{d}\cos\theta_{\ell}\,\mathrm{d}\cos\theta_{K}\,\mathrm{d}\hat{\phi}}=\frac{9}{16\pi}&\left[\frac{}{}F_{\rm
L}\cos^{2}\theta_{K}+\frac{3}{4}(1-F_{\rm
L})(1-\cos^{2}\theta_{K})~{}~{}-\right.\\\ &\left.~{}\frac{}{}\,F_{\rm
L}\cos^{2}\theta_{K}(2\cos^{2}\theta_{\ell}-1)~{}~{}+\right.\\\
&\left.~{}\frac{}{}~{}\frac{1}{4}(1-F_{\rm
L})(1-\cos^{2}\theta_{K})(2\cos^{2}\theta_{\ell}-1)~{}~{}+\right.\\\
&\left.~{}\frac{}{}~{}S_{3}(1-\cos^{2}\theta_{K})(1-\cos^{2}\theta_{\ell})\cos
2\hat{\phi}~{}~{}+\right.\\\ &\left.~{}\frac{}{}~{}\frac{4}{3}A_{\rm
FB}(1-\cos^{2}\theta_{K})\cos\theta_{\ell}~{}~{}+\right.\\\
&\left.~{}\frac{}{}~{}A_{9}(1-\cos^{2}\theta_{K})(1-\cos^{2}\theta_{\ell})\sin
2\hat{\phi}\frac{}{}~{}\right]~{}.\end{split}$ (4)

This expression involves the same set of observables that can be extracted
from fits to the one-dimensional angular projections.

At large recoil it is also advantageous to reformulate Eq. 4 in terms of the
observables $A_{\rm T}^{2}$ and $A_{\rm T}^{\rm Re}$, where $A_{\rm
FB}=\frac{3}{4}\left(1-F_{\rm L}\right)A_{\rm T}^{\rm Re}$. These so called
“transverse” observables only depend on a subset of the decay amplitudes (with
transverse polarisation of the $K^{*0}$) and are expected to come with reduced
form-factor uncertainties [4, 9]. A first measurement of $A_{\rm T}^{2}$ was
performed by the CDF experiment [8].

This paper presents a measurement of the differential branching fraction
($\mathrm{d}{\cal B}/\mathrm{d}q^{2}$), $A_{\rm FB}$, $F_{\rm L}$, $S_{3}$ and
$A_{9}$ of the $B^{0}\\!\rightarrow K^{*0}\mu^{+}\mu^{-}$ decay in six bins of
$q^{2}$. Measurements of the transverse observables $A_{\rm T}^{2}$ and
$A_{\rm T}^{\rm Re}$ are also presented. The analysis is based on a dataset,
corresponding to 1.0$\mbox{\,fb}^{-1}$ of integrated luminosity, collected by
the LHCb detector in $\sqrt{s}=7\mathrm{\,Te\kern-1.00006ptV}$ $pp$ collisions
in 2011. Section 2 describes the experimental setup used in the analyses.
Section 3 describes the event selection. Section 4 discusses potential sources
of peaking background. Section 5 describes the treatment of the detector
acceptance in the analysis. Section 6 discusses the measurement of
$\mathrm{d}{\cal B}/\mathrm{d}q^{2}$. The angular analysis of the decay, in
terms of $\cos\theta_{\ell}$, $\cos\theta_{K}$ and $\hat{\phi}$, is described
in Sec. 7. Finally, a first measurement of the zero-crossing point of $A_{\rm
FB}$ is presented in Sec. 8.

##  2 The LHCb detector

The LHCb detector [10] is a single-arm forward spectrometer, covering the
pseudorapidity range $2<\eta<5$, that is designed to study $b$ and $c$ hadron
decays. A dipole magnet with a bending power of 4 Tm and a large area tracking
detector provide momentum resolution ranging from 0.4% for tracks with a
momentum of 5${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ to 0.6% for a momentum of
100${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$. A silicon microstrip detector,
located around the $pp$ interaction region, provides excellent separation of
$B$ meson decay vertices from the primary $pp$ interaction and impact
parameter resolution of 20$\,\upmu\rm m$ for tracks with high transverse
momentum ($p_{\rm T}$). Two ring-imaging Cherenkov (RICH) detectors [11]
provide kaon-pion separation in the momentum range
$2-100{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$. Muons are identified based on
hits created in a system of multiwire proportional chambers interleaved with
layers of iron. The LHCb trigger [12] comprises a hardware trigger and a two-
stage software trigger that performs a full event reconstruction.

Samples of simulated events are used to estimate the contribution from
specific sources of exclusive backgrounds and the efficiency to trigger,
reconstruct and select the $B^{0}\\!\rightarrow K^{*0}\mu^{+}\mu^{-}$ signal.
The simulated $pp$ interactions are generated using Pythia 6.4 [13] with a
specific LHCb configuration [14]. Decays of hadronic particles are then
described by EvtGen [15] in which final state radiation is generated using
Photos [16]. Finally, the Geant4 toolkit [17, *Agostinelli:2002hh] is used to
simulate the detector response to the particles produced by Pythia/EvtGen, as
described in Ref. [19]. The simulated samples are corrected for known
differences between data and simulation in the $B^{0}$ momentum spectrum, the
detector impact parameter resolution, particle identification [11] and
tracking system performance using control samples from the data.

##  3 Selection of signal candidates

The $B^{0}\\!\rightarrow K^{*0}\mu^{+}\mu^{-}$ candidates are selected from
events that have been triggered by a muon with $\mbox{$p_{\rm
T}$}>1.5{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$, in the hardware trigger. In the
first stage of the software trigger, candidates are selected if there is a
reconstructed track in the event with high impact parameter ($>125\,\upmu\rm
m$) with respect to one of the primary $pp$ interactions and $\mbox{$p_{\rm
T}$}>1.5{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$. In the second stage of the
software trigger, candidates are triggered on the kinematic properties of the
partially or fully reconstructed $B^{0}$ candidate [12].

Signal candidates are then required to pass a set of loose (pre-)selection
requirements. Candidates are selected for further analysis if: the $B^{0}$
decay vertex is separated from the primary $pp$ interaction; the $B^{0}$
candidate impact parameter is small, and the impact parameters of the charged
kaon, pion and muons are large, with respect to the primary $pp$ interaction;
and the angle between the $B^{0}$ momentum vector and the vector between the
primary $pp$ interaction and the $B^{0}$ decay vertex is small. Candidates are
retained if their $K^{+}\pi^{-}$ invariant mass is in the range
$792<m({K^{+}\pi^{-}})<992{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$.

A multivariate selection, using a boosted decision tree (BDT) [20] with the
AdaBoost algorithm[21], is applied to further reduce the level of
combinatorial background. The BDT is identical to that described in Ref. [5].
It has been trained on a data sample, corresponding to 36$\mbox{\,pb}^{-1}$ of
integrated luminosity, collected by the LHCb experiment in 2010. A sample of
$B^{0}\\!\rightarrow K^{*0}{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$
(${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\\!\rightarrow\mu^{+}\mu^{-}$)
candidates is used to represent the $B^{0}\\!\rightarrow K^{*0}\mu^{+}\mu^{-}$
signal in the BDT training. The decay $B^{0}\\!\rightarrow
K^{*0}{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ is used throughout this
analysis as a control channel. Candidates from the $B^{0}\\!\rightarrow
K^{*0}\mu^{+}\mu^{-}$ upper mass sideband
($5350<m(K^{+}\pi^{-}\mu^{+}\mu^{-})<5600{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$)
are used as a background sample. Candidates with invariant masses below the
nominal $B^{0}$ mass contain a significant contribution from partially
reconstructed $B$ decays and are not used in the BDT training or in the
subsequent analysis. They are removed by requiring that candidates have
$m({K^{+}\pi^{-}\mu^{+}\mu^{-}})>5150{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$.
The BDT uses predominantly geometric variables, including the variables used
in the above pre-selection. It also includes information on the quality of the
$B^{0}$ vertex and the fit $\chi^{2}$ of the four tracks. Finally the BDT
includes information from the RICH and muon systems on the likelihood that the
kaon, pion and muons are correctly identified. Care has been taken to ensure
that the BDT does not preferentially select regions of $q^{2}$,
$K^{+}\pi^{-}\mu^{+}\mu^{-}$ invariant mass or of the
$K^{+}\pi^{-}\mu^{+}\mu^{-}$ angular distribution. The multivariate selection
retains 78% of the signal and 12% of the background that remains after the
pre-selection.

Figure 1: Distribution of $\mu^{+}\mu^{-}$ versus $K^{+}\pi^{-}\mu^{+}\mu^{-}$
invariant mass of selected $B^{0}\\!\rightarrow K^{*0}\mu^{+}\mu^{-}$
candidates. The vertical lines indicate a $\pm
50{\mathrm{\,Me\kern-0.90005ptV\\!/}c^{2}}$ signal mass window around the
nominal $B^{0}$ mass. The horizontal lines indicate the two veto regions that
are used to remove ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ and
$\psi{(2S)}\\!\rightarrow\mu^{+}\mu^{-}$ decays. The $B^{0}\\!\rightarrow
K^{*0}\mu^{+}\mu^{-}$ signal is clearly visible outside of the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ and
$\psi{(2S)}\\!\rightarrow\mu^{+}\mu^{-}$ windows.

Figure 1 shows the $\mu^{+}\mu^{-}$ versus $K^{+}\pi^{-}\mu^{+}\mu^{-}$
invariant mass of the selected candidates. The $B^{0}\\!\rightarrow
K^{*0}\mu^{+}\mu^{-}$ signal, which peaks in $K^{+}\pi^{-}\mu^{+}\mu^{-}$
invariant mass, and populates the full range of the dimuon invariant mass
range, is clearly visible.

##  4 Exclusive and partially reconstructed backgrounds

Several sources of peaking background have been studied using samples of
simulated events, corrected to reflect the difference in particle
identification (and misidentification) performance between the data and
simulation. Sources of background that are not reduced to a negligible level
by the pre- and multivariate-selections are described below.

The decays $B^{0}\\!\rightarrow K^{*0}{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}$ and $B^{0}\\!\rightarrow K^{*0}\psi{(2S)}$, where
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ and
$\psi{(2S)}\\!\rightarrow\mu^{+}\mu^{-}$, are removed by rejecting candidates
with $2946<m({\mu^{+}\mu^{-}})<3176{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$
and $3586<m({\mu^{+}\mu^{-}})<3766{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$.
These vetoes are extended downwards by
150${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ in $m({\mu^{+}\mu^{-}})$ for
$B^{0}\\!\rightarrow K^{*0}\mu^{+}\mu^{-}$ candidates with masses
$5150<m({K^{+}\pi^{-}\mu^{+}\mu^{-}})<5230{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$
to account for the radiative tails of the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ and $\psi{(2S)}$ mesons. They
are also extended upwards by 25${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ for
candidates with masses above the $B^{0}$ mass to account for the small
percentage of ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ or $\psi{(2S)}$
decays that are misreconstructed at higher masses. The
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ and $\psi{(2S)}$ vetoes are
shown in Fig. 1.

The decay $B^{0}\\!\rightarrow K^{*0}{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}$ can also form a source of peaking background if the kaon or pion is
misidentified as a muon and swapped with one of the muons from the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ decay. This background is
removed by rejecting candidates that have a $K^{+}\mu^{-}$ or $\pi^{-}\mu^{+}$
invariant mass (where the kaon or pion is assigned the muon mass) in the range
$3036<m({\mu^{+}\mu^{-}})<3156{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ if the
kaon or pion can also be matched to hits in the muon stations. A similar veto
is applied for the decay $B^{0}\\!\rightarrow K^{*0}\psi{(2S)}$.

The decay $B^{0}_{s}\\!\rightarrow\phi\mu^{+}\mu^{-}$, where
$\phi\\!\rightarrow K^{+}K^{-}$, is removed by rejecting candidates if the
$K^{+}\pi^{-}$ mass is consistent with originating from a $\phi\\!\rightarrow
K^{+}K^{-}$ decay and the pion is kaon-like according to the RICH detectors. A
similar veto is applied to remove $\mathchar
28931\relax^{0}_{b}\\!\rightarrow\mathchar
28931\relax^{*}(1520)\mu^{+}\mu^{-}$ ($\mathchar
28931\relax^{*}(1520)\\!\rightarrow pK^{-}$) decays.

There is also a source of background from the decay $B^{+}\\!\rightarrow
K^{+}\mu^{+}\mu^{-}$ that appears in the upper mass sideband and has a peaking
structure in $\cos\theta_{K}$. This background arises when a $K^{*0}$
candidate is formed using a pion from the other $B$ decay in the event, and is
removed by vetoing events that have a $K^{+}\mu^{+}\mu^{-}$ invariant mass in
the range
$5230<m({K^{+}\mu^{+}\mu^{-}})<5330{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$.
The fraction of combinatorial background candidates removed by this veto is
small.

After these selection requirements the dominant sources of peaking background
are expected to be from the decays $B^{0}\\!\rightarrow
K^{*0}{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ (where the kaon or pion is
misidentified as a muon and a muon as a pion or kaon),
$B^{0}_{s}\\!\rightarrow\phi$$\mu^{+}\mu^{-}$ and $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\\!\rightarrow
K^{*0}\mu^{+}\mu^{-}$ at the levels of $(0.3\pm 0.1)\%$, $(1.2\pm 0.5)\%$ and
$(1.0\pm 1.0)\%$, respectively. The rate of the decay $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\\!\rightarrow
K^{*0}\mu^{+}\mu^{-}$ is estimated using the fragmentation fraction
$f_{s}/f_{d}$ [22] and assuming the branching fraction of this decay is
suppressed by the ratio of CKM elements $|V_{td}/V_{ts}|^{2}$ with respect to
$B^{0}\\!\rightarrow K^{*0}\mu^{+}\mu^{-}$. To estimate the systematic
uncertainty arising from the assumed $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\\!\rightarrow
K^{*0}\mu^{+}\mu^{-}$ signal, the expectation is varied by 100%. Finally, the
probability for a decay $B^{0}\\!\rightarrow K^{*0}\mu^{+}\mu^{-}$ to be
misidentified as $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\\!\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}\mu^{+}\mu^{-}$ is estimated to be
$(0.85\pm 0.02)\%$ using simulated events.

##  5 Detector acceptance and selection biases

The geometrical acceptance of the detector, the trigger, the event
reconstruction and selection can all bias the angular distribution of the
selected candidates. At low $q^{2}$ there are large distortions of the angular
distribution at extreme values of $\cos\theta_{\ell}$
($|\cos\theta_{\ell}|\sim 1$). These arise from the requirement that muons
have momentum $p{~{}\raise 1.49994pt\hbox{$>$}\kern-8.50006pt\lower
3.50006pt\hbox{$\sim$}~{}}3{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ to traverse
the LHCb muon system. Distortions are also visible in the $\cos\theta_{K}$
angular distribution. They arise from the momentum needed for a track to reach
the tracking system downstream of the dipole magnet, and from the impact
parameter requirements in the pre-selection. The acceptance in
$\cos\theta_{K}$ is asymmetric due to the momentum imbalance between the pion
and kaon from the $K^{*0}$ decay in the laboratory frame (due to the boost).

Acceptance effects are accounted for, in a model-independent way by weighting
candidates by the inverse of their efficiency determined from simulation. The
event weighting takes into account the variation of the acceptance in $q^{2}$
to give an unbiased estimate of the observables over the $q^{2}$ bin. The
candidate weights are normalised such that they have mean 1.0. The resulting
distribution of weights in each $q^{2}$ bin has a root-mean-square in the
range $0.2-0.4$. Less than 2% of the candidates have weights larger than 2.0.

The weights are determined using a large sample of simulated three-body
$B^{0}\\!\rightarrow K^{*0}\mu^{+}\mu^{-}$ phase-space decays. They are
determined separately in fine bins of $q^{2}$ with widths:
$0.1\mathrm{\,Ge\kern-1.00006ptV}^{2}/c^{4}$ for
$q^{2}<1\mathrm{\,Ge\kern-1.00006ptV}^{2}/c^{4}$;
$0.2\mathrm{\,Ge\kern-1.00006ptV}^{2}/c^{4}$ in the range
$1<q^{2}<6\mathrm{\,Ge\kern-1.00006ptV}^{2}/c^{4}$; and
$0.5\mathrm{\,Ge\kern-1.00006ptV}^{2}/c^{4}$ for
$q^{2}>6\mathrm{\,Ge\kern-1.00006ptV}^{2}/c^{4}$. The width of the $q^{2}$
bins is motivated by the size of the simulated sample and by the rate of
variation of the acceptance in $q^{2}$. Inside the $q^{2}$ bins, the angular
acceptance is assumed to factorise such that
$\varepsilon(\cos\theta_{\ell},\cos\theta_{K},\phi)=\varepsilon(\cos\theta_{\ell})\varepsilon(\cos\theta_{K})\varepsilon(\phi)$.
This factorisation is validated at the level of 5% in the phase-space sample.
The treatment of the event weights is discussed in more detail in Sec. 7.1,
when determining the statistical uncertainty on the angular observables.

Event weights are also used to account for the fraction of background
candidates that were removed in the lower mass
($m(K^{+}\pi^{-}\mu^{+}\mu^{-})<5230{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$)
and upper mass
($m(K^{+}\pi^{-}\mu^{+}\mu^{-})>5330{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$)
sidebands by the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ and
$\psi{(2S)}$ vetoes described in Sec. 4 (and shown in Fig. 1). In each $q^{2}$
bin, a linear extrapolation in $q^{2}$ is used to estimate this fraction and
the resulting event weights.

##  6 Differential branching fraction

The angular and differential branching fraction analyses are performed in six
bins of $q^{2}$, which are the same as those used in Ref. [7]. The
$K^{+}\pi^{-}\mu^{+}\mu^{-}$ invariant mass distribution of candidates in
these $q^{2}$ bins is shown in Fig. 2.

The number of signal candidates in each of the $q^{2}$ bins is estimated by
performing an extended unbinned maximum likelihood fit to the
$K^{+}\pi^{-}\mu^{+}\mu^{-}$ invariant mass distribution. The signal shape is
taken from a fit to the $B^{0}\\!\rightarrow
K^{*0}{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ control sample and is
parameterised by the sum of two Crystal Ball [23] functions that differ only
by the width of the Gaussian component. The combinatorial background is
described by an exponential distribution. The decay $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\\!\rightarrow
K^{*0}\mu^{+}\mu^{-}$, which forms a peaking background, is assumed to have a
shape identical to that of the $B^{0}\\!\rightarrow K^{*0}\mu^{+}\mu^{-}$
signal, but shifted in mass by the $B^{0}_{s}-B^{0}$ mass difference [24].
Contributions from the decays $B^{0}_{s}\\!\rightarrow\phi\mu^{+}\mu^{-}$ and
$B^{0}\\!\rightarrow K^{*0}{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$
(where the $\mu^{-}$ is swapped with the $\pi^{-}$) are also included. The
shapes of these backgrounds are taken from samples of simulated events. The
sizes of the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\\!\rightarrow
K^{*0}\mu^{+}\mu^{-}$, $B^{0}_{s}\\!\rightarrow\phi\mu^{+}\mu^{-}$ and
$B^{0}\\!\rightarrow K^{*0}{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$
backgrounds are fixed with respect to the fitted $B^{0}\\!\rightarrow
K^{*0}\mu^{+}\mu^{-}$ signal yield according to the ratios described in Sec.
4. These backgrounds are varied to evaluate the corresponding systematic
uncertainty. The resulting signal yields are given in Table 1. In the full
$0.1<q^{2}<19.0\mathrm{\,Ge\kern-1.00006ptV}^{2}/c^{4}$ range, the fit yields
$883\pm 34$ signal decays.

Figure 2: Invariant mass distributions of $K^{+}\pi^{-}\mu^{+}\mu^{-}$
candidates in the six $q^{2}$ bins used in the analysis. The candidates have
been weighted to account for the detector acceptance (see text). Contributions
from exclusive (peaking) backgrounds are negligible after applying the vetoes
described in Sec. 4.

The differential branching fraction of the decay $B^{0}\\!\rightarrow
K^{*0}\mu^{+}\mu^{-}$, in each $q^{2}$ bin, is estimated by normalising the
$B^{0}\\!\rightarrow K^{*0}\mu^{+}\mu^{-}$ yield, $N_{\text{sig}}$, to the
total event yield of the $B^{0}\\!\rightarrow
K^{*0}{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ control sample,
$N_{K^{*0}{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$, and correcting for
the relative efficiency between the two decays,
$\varepsilon_{K^{*0}{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}/\varepsilon_{K^{*0}\mu^{+}\mu^{-}}$,

$\frac{\mathrm{d}{\cal
B}}{\mathrm{d}q^{2}}=\frac{1}{q^{2}_{\text{max}}-q^{2}_{\text{min}}}\frac{N_{\text{sig}}}{N_{K^{*0}{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}}\frac{\varepsilon_{K^{*0}{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}}{\varepsilon_{K^{*0}\mu^{+}\mu^{-}}}\times{\cal B}(B^{0}\\!\rightarrow
K^{*0}{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu})\times{\cal
B}({J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\\!\rightarrow\mu^{+}\mu^{-})~{}.$ (5)

The branching fractions ${\cal B}(B^{0}\\!\rightarrow
K^{*0}{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu})$ and ${\cal
B}({J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\\!\rightarrow\mu^{+}\mu^{-})$
are $(1.31\pm 0.03\pm 0.08)\times 10^{-3}$ [25] and $(5.93\pm 0.06)\times
10^{-2}$ [24], respectively.

The efficiency ratio,
$\varepsilon_{K^{*0}{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}/\varepsilon_{K^{*0}\mu^{+}\mu^{-}}$, depends on the unknown angular
distribution of the $B^{0}\\!\rightarrow K^{*0}\mu^{+}\mu^{-}$ decay. To avoid
making any assumption on the angular distribution, the event-by-event weights
described in Sec. 5 are used to estimate the average efficiency of the
$B^{0}\\!\rightarrow K^{*0}{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$
candidates and the signal candidates in each $q^{2}$ bin.

###  6.1 Comparison with theory

The resulting differential branching fraction of the decay
$B^{0}\\!\rightarrow K^{*0}\mu^{+}\mu^{-}$ is shown in Fig. 3 and in Table 1.
The bands shown in Fig. 3 indicate the theoretical prediction for the
differential branching fraction. The calculation of the bands is described in
Ref. [26].222A consistent set of SM predictions, averaged over each $q^{2}$
bin, have recently also been provided by the authors of Ref. [27].  In the low
$q^{2}$ region, the calculations are based on QCD factorisation and soft
collinear effective theory (SCET) [28], which profit from having a heavy
$B^{0}$ meson and an energetic $K^{*0}$ meson. In the soft-recoil, high
$q^{2}$ region, an operator product expansion in inverse $b$-quark mass
($1/m_{b}$) and $1/\sqrt{q^{2}}$ is used to estimate the long-distance
contributions from quark loops [29, 30]. No theory prediction is included in
the region close to the narrow $c\overline{}c$ resonances (the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ and $\psi{(2S)}$) where the
assumptions from QCD factorisation, SCET and the operator product expansion
break down. The treatment of this region is discussed in Ref. [31]. The form-
factor calculations are taken from Ref. [32]. A dimensional estimate is made
of the uncertainty on the decay amplitudes from QCD factorisation and SCET of
$\mathcal{O}(\Lambda_{\text{QCD}}/m_{b})$ [33]. Contributions from light-quark
resonances at large recoil (low $q^{2}$) have been neglected. A discussion of
these contributions can be found in Ref. [34]. The same techniques are
employed in calculations of the angular observables described in Sec. 7.

Table 1: Signal yield ($N_{\text{sig}}$) and differential branching fraction ($\mathrm{d}{\cal B}/\mathrm{d}q^{2}$) of the $B^{0}\\!\rightarrow K^{*0}\mu^{+}\mu^{-}$ decay in the six $q^{2}$ bins used in this analysis. Results are also presented in the $1<q^{2}<6\mathrm{\,Ge\kern-0.90005ptV}^{2}/c^{4}$ range where theoretical uncertainties are best controlled. The first and second uncertainties are statistical and systematic. The third uncertainty comes from the uncertainty on the $B^{0}\\!\rightarrow K^{*0}{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ and ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\\!\rightarrow\mu^{+}\mu^{-}$ branching fractions. The final uncertainty on $\mathrm{d}{\cal B}/\mathrm{d}q^{2}$ comes from an estimate of the pollution from non-$K^{*0}$ $B^{0}\\!\rightarrow K^{+}\pi^{-}\mu^{+}\mu^{-}$ decays in the $792<m({K^{+}\pi^{-}})<992{\mathrm{\,Me\kern-0.90005ptV\\!/}c^{2}}$ mass window (see Sec. 7.3.2). $q^{2}$ $(\mathrm{\,Ge\kern-1.00006ptV}^{2}/c^{4})$ | $N_{\text{sig}}$ |  $\mathrm{d}{\cal B}/\mathrm{d}q^{2}$ $(10^{-7}\mathrm{\,Ge\kern-1.00006ptV}^{-2}c^{4})$
---|---|---
$\phantom{0}0.10-\phantom{0}2.00$ | $140\pm 13$ | $0.60\pm 0.06\pm 0.05\pm 0.04\,^{+0.00}_{-0.05}$
$\phantom{0}2.00-\phantom{0}4.30$ | $\phantom{0}73\pm 11$ | $0.30\pm 0.03\pm 0.03\pm 0.02\,^{+0.00}_{-0.02}$
$\phantom{0}4.30-\phantom{0}8.68$ | $271\pm 19$ | $0.49\pm 0.04\pm 0.04\pm 0.03\,^{+0.00}_{-0.04}$
$10.09-12.86$ | $168\pm 15$ | $0.43\pm 0.04\pm 0.04\pm 0.03\,^{+0.00}_{-0.03}$
$14.18-16.00$ | $115\pm 12$ | $0.56\pm 0.06\pm 0.04\pm 0.04\,^{+0.00}_{-0.05}$
$16.00-19.00$ | $116\pm 13$ | $0.41\pm 0.04\pm 0.04\pm 0.03\,^{+0.00}_{-0.03}$
$\phantom{0}1.00-\phantom{0}6.00$ | $197\pm 17$ | $0.34\pm 0.03\pm 0.04\pm 0.02\,^{+0.00}_{-0.03}$
Figure 3: Differential branching fraction of the $B^{0}\\!\rightarrow
K^{*0}\mu^{+}\mu^{-}$ decay as a function of the dimuon invariant mass
squared. The data are overlaid with a SM prediction (see text) for the decay
(light-blue band). A rate average of the SM prediction across each $q^{2}$ bin
is indicated by the dark (purple) rectangular regions. No SM prediction is
included in the region close to the narrow $c\overline{}c$ resonances.

###  6.2 Systematic uncertainty

The largest sources of systematic uncertainty on the $B^{0}\\!\rightarrow
K^{*0}\mu^{+}\mu^{-}$ differential branching fraction come from the $\sim 6\%$
uncertainty on the combined $B^{0}\\!\rightarrow
K^{*0}{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ and
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\\!\rightarrow\mu^{+}\mu^{-}$
branching fractions and from the uncertainty on the pollution of non-$K^{*0}$
decays in the
$792<m({K^{+}\pi^{-}})<992{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ mass
window. The latter pollution arises from decays where the $K^{+}\pi^{-}$
system is in an S- rather than P-wave configuration. For the decay
$B^{0}\\!\rightarrow K^{*0}{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$, the
S-wave pollution is known to be at the level of a few percent [35]. The effect
of S-wave pollution on the decay $B^{0}\\!\rightarrow K^{*0}\mu^{+}\mu^{-}$ is
considered in Sec. 7.3.2. No S-wave correction needs to be applied to the
yield of $B^{0}\\!\rightarrow K^{*0}{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}$ decays in the present analysis, since the branching fraction used in
the normalisation (from Ref. [25]) corresponds to a measurement of the decay
$B^{0}\\!\rightarrow K^{+}\pi^{-}{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$
over the same $m(K^{+}\pi^{-})$ window used in this analysis.

The uncertainty associated with the data-derived corrections to the
simulation, which were described in Sec. 2, is estimated to be $1-2\%$.
Varying the level of the peaking backgrounds within their uncertainties
changes the differential branching fraction by 1% and this variation is taken
as a systematic uncertainty. In the simulation a small variation in the
$K^{+}\pi^{-}\mu^{+}\mu^{-}$ invariant mass resolution is seen between
$B^{0}\\!\rightarrow K^{*0}{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ and
$B^{0}\\!\rightarrow K^{*0}\mu^{+}\mu^{-}$ decays at low and high $q^{2}$, due
to differences in the decay kinematics. The maximum size of this variation in
the simulation is 5%. A conservative systematic uncertainty is assigned by
varying the mass resolution of the signal decay by this amount in every
$q^{2}$ bin and taking the deviation from the nominal fit as the uncertainty.

##  7 Angular analysis

This section describes the analysis of the $\cos\theta_{\ell}$,
$\cos\theta_{K}$ and $\hat{\phi}$ distribution after applying the
transformations that were described earlier. These transformations reduce the
full angular distribution from 11 angular terms to one that only depends on
four observables: $A_{\rm FB}$, $F_{\rm L}$, $S_{3}$ and $A_{9}$. The
resulting angular distribution is given in Eq. 4 in Sec. 1.

In order for Eq. 4 to remain positive in all regions of the allowed phase
space, the observables $A_{\rm FB}$, $F_{\rm L}$, $S_{3}$ and $A_{9}$ must
satisfy the constraints

$|A_{\rm FB}|\leq\frac{3}{4}(1-F_{\rm
L})~{},~{}|A_{9}|\leq\frac{1}{2}(1-F_{\rm
L})~{}~{}\text{and}~{}~{}|S_{3}|\leq\frac{1}{2}(1-F_{\rm L})~{}.$

These requirements are automatically taken into account if $A_{\rm FB}$ and
$S_{3}$ are replaced by the theoretically cleaner transverse observables,
$A_{\rm T}^{\rm Re}$ and $A_{\rm T}^{2}$,

$A_{\rm FB}=\frac{3}{4}(1-F_{\rm L})A_{\rm T}^{\rm
Re}~{}~{}\text{and}~{}~{}S_{3}=\frac{1}{2}(1-F_{\rm L})A_{\rm T}^{2}~{},$

which are defined in the range $[-1,1]$.

In each of the $q^{2}$ bins, $A_{\rm FB}$ ($A_{\rm T}^{\rm Re}$), $F_{\rm L}$,
$S_{3}$ ($A_{\rm T}^{2}$) and $A_{9}$ are estimated by performing an unbinned
maximum likelihood fit to the $\cos\theta_{\ell}$, $\cos\theta_{K}$ and
$\hat{\phi}$ distributions of the $B^{0}\\!\rightarrow K^{*0}\mu^{+}\mu^{-}$
candidates. The $K^{+}\pi^{-}\mu^{+}\mu^{-}$ invariant mass of the candidates
is also included in the fit to separate between signal- and background-like
candidates. The background angular distribution is described using the product
of three second-order Chebychev polynomials under the assumption that the
background can be factorised into three single angle distributions. This
assumption has been validated on the data sidebands
($5350<m({K^{+}\pi^{-}\mu^{+}\mu^{-}})<5600{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$).
A dilution factor ($\mathcal{D}=1-2\omega$) is included in the likelihood fit
for $A_{\rm FB}$ and $A_{9}$, to account at first order for the small
probability ($\omega$) for a decay $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\\!\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}\mu^{+}\mu^{-}$ to be misidentified
as $B^{0}\\!\rightarrow K^{*0}\mu^{+}\mu^{-}$. The value of $\omega$ is fixed
to $0.85\%$ in the fit (see Sec. 4).

Two fits to the dataset are performed: one, with the signal angular
distribution described by Eq. 4, to measure $F_{\rm L}$, $A_{\rm FB}$, $S_{3}$
and $A_{9}$ and a second replacing $A_{\rm FB}$ and $S_{3}$ with the
observables $A_{\rm T}^{\rm Re}$ and $A_{\rm T}^{2}$. The angular observables
vary with $q^{2}$ within the $q^{2}$ bins used in the analysis. The measured
quantities therefore correspond to averages over these $q^{2}$ bins. For the
transverse observables, where the observable appears alongside $1-F_{\rm L}$
in the angular distribution, the averaging is complicated by the $q^{2}$
dependence of both the observable and $F_{\rm L}$. In this case, the measured
quantity corresponds to a weighted average of the transverse observable over
$q^{2}$, with a weight $(1-F_{\rm L})\mathrm{d}\Gamma/\mathrm{d}q^{2}$.

###  7.1 Statistical uncertainty on the angular observables

The results of the angular fits are presented in Table 2 and in Figs. 4 and 5.
The 68% confidence intervals are estimated using pseudo-experiments and the
Feldman-Cousins technique [36].333Nuisance parameters are treated according to
the “plug-in” method (see, for example, Ref. [37]). This avoids any potential
bias on the parameter uncertainty that could have otherwise come from using
event weights in the likelihood fit or from boundary issues arising in the
fitting. The observables are each treated separately in this procedure. For
example, when determining the interval on $A_{\rm FB}$, the observables
$F_{\rm L}$, $S_{3}$ and $A_{9}$ are treated as if they were nuisance
parameters. At each value of the angular observable being considered, the
maximum likelihood estimate of the nuisance parameters (which also include the
background parameters) is used when generating the pseudo-experiments. The
resulting confidence intervals do not express correlations between the
different observables. The treatment of systematic uncertainties on the
angular observables is described in Sec. 7.3.

The final column of Table 2 contains the p-value of the SM point in each
$q^{2}$ bin, which is defined as the probability to observe a difference
between the log-likelihood of the SM point compared to the best fit point
larger than that seen in the data. They are estimated in a similar way to the
Feldman-Cousins intervals by: generating a large ensemble of pseudo-
experiments, with all of the angular observables fixed to the central value of
the SM prediction; and performing two fits to the pseudo-experiments, one with
all of the angular observables fixed to their SM values and one varying them
freely. The data are then fitted in a similar manner and the p-value estimated
by comparing the ratio of likelihoods obtained for the data to those of the
pseudo-experiments. The p-values lie in the range $0.18-0.72$ and indicate
good agreement with the SM hypothesis.

As a cross-check, a third fit is also performed in which the sign of the angle
$\phi$ for $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ decays is
flipped to measure $S_{9}$ in place of $A_{9}$ in the angular distribution.
The term $S_{9}$ is expected to be suppressed by the size of the strong phases
and be close to zero in every $q^{2}$ bin. $A_{\rm FB}$ has also been cross-
checked by performing a counting experiment in bins of $q^{2}$. A consistent
result is obtained in every bin.

Table 2: Fraction of longitudinal polarisation of the $K^{*0}$, $F_{\rm L}$, dimuon system forward-backward asymmetry, $A_{\rm FB}$ and the angular observables $S_{3}$, $S_{9}$ and $A_{9}$ from the $B^{0}\\!\rightarrow K^{*0}\mu^{+}\mu^{-}$ decay in the six bins of dimuon invariant mass squared, $q^{2}$, used in the analysis. The lower table includes the transverse observables $A_{\rm T}^{\rm Re}$ and $A_{\rm T}^{2}$, which have reduced form-factor uncertainties. Results are also presented in the $1<q^{2}<6\mathrm{\,Ge\kern-0.90005ptV}^{2}/c^{4}$ range where theoretical uncertainties are best controlled. In the large-recoil bin, $0.1<q^{2}<2.0\mathrm{\,Ge\kern-0.90005ptV}^{2}/c^{4}$, two results are given to highlight the size of the correction needed to account for changes in the angular distribution that occur when $q^{2}{~{}\raise 1.34995pt\hbox{$<$}\kern-7.65005pt\lower 3.15005pt\hbox{$\sim$}~{}}1\mathrm{\,Ge\kern-0.90005ptV}^{2}/c^{4}$ (see Sec. 7.2). The value of $F_{\rm L}$ is independent of this correction. The final column contains the p-value for the SM point (see text). No SM prediction, and consequently no p-value, is available for the $10.09<q^{2}<12.86\mathrm{\,Ge\kern-0.90005ptV}^{2}/c^{4}$ range. $q^{2}$ $(\mathrm{\,Ge\kern-1.00006ptV}^{2}/c^{4})$ | $F_{\rm L}$ | $A_{\rm FB}$ | $S_{3}$ | $S_{9}$
---|---|---|---|---
$0.10-2.00$ | $0.37\,^{+0.10}_{-0.09}\,{}^{+0.04}_{-0.03}$ | $-0.02\,^{+0.12}_{-0.12}\,{}^{+0.01}_{-0.01}$ | $-0.04\,^{+0.10}_{-0.10}\,{}^{+0.01}_{-0.01}$ | $\phantom{-}0.05\,^{+0.10}_{-0.09}\,{}^{+0.01}_{-0.01}$
(uncorrected) |  |  |  |
$0.10-2.00$ | $0.37\,^{+0.10}_{-0.09}\,{}^{+0.04}_{-0.03}$ | $-0.02\,_{-0.13}^{+0.13}\,{}_{-0.01}^{+0.01}$ | $-0.05\,_{-0.12}^{+0.12}\,{}_{-0.01}^{+0.01}$ | $\phantom{-}0.06\,_{-0.12}^{+0.12}\,{}_{-0.01}^{+0.01}$
(corrected) |  |  |  |
$2.00-4.30$ | $0.74\,^{+0.10}_{-0.09}\,{}^{+0.02}_{-0.03}$ | $-0.20\,^{+0.08}_{-0.08}\,{}^{+0.01}_{-0.01}$ | $-0.04\,^{+0.10}_{-0.06}\,{}^{+0.01}_{-0.01}$ | $-0.03\,^{+0.11}_{-0.04}\,{}^{+0.01}_{-0.01}$
$4.30-8.68$ | $0.57\,^{+0.07}_{-0.07}\,{}^{+0.03}_{-0.03}$ | $\phantom{-}0.16\,^{+0.06}_{-0.05}\,{}^{+0.01}_{-0.01}$ | $\phantom{-}0.08\,^{+0.07}_{-0.06}\,{}^{+0.01}_{-0.01}$ | $\phantom{-}0.01\,^{+0.07}_{-0.08}\,{}^{+0.01}_{-0.01}$
$10.09-12.86$ | $0.48\,^{+0.08}_{-0.09}\,{}^{+0.03}_{-0.03}$ | $\phantom{-}0.28\,^{+0.07}_{-0.06}\,{}^{+0.02}_{-0.02}$ | $-0.16\,^{+0.11}_{-0.07}\,{}^{+0.01}_{-0.01}$ | $-0.01\,^{+0.10}_{-0.11}\,{}^{+0.01}_{-0.01}$
$14.18-16.00$ | $0.33\,^{+0.08}_{-0.07}\,{}^{+0.02}_{-0.03}$ | $\phantom{-}0.51\,^{+0.07}_{-0.05}\,{}^{+0.02}_{-0.02}$ | $\phantom{-}0.03\,^{+0.09}_{-0.10}\,{}^{+0.01}_{-0.01}$ | $\phantom{-}0.00\,^{+0.09}_{-0.08}\,{}^{+0.01}_{-0.01}$
$16.00-19.00$ | $0.38\,^{+0.09}_{-0.07}\,{}^{+0.03}_{-0.03}$ | $\phantom{-}0.30\,^{+0.08}_{-0.08}\,{}^{+0.01}_{-0.02}$ | $-0.22\,^{+0.10}_{-0.09}\,{}^{+0.02}_{-0.01}$ | $\phantom{-}0.06\,^{+0.11}_{-0.10}\,{}^{+0.01}_{-0.01}$
$1.00-6.00$ | $0.65\,^{+0.08}_{-0.07}\,{}^{+0.03}_{-0.03}$ | $-0.17\,^{+0.06}_{-0.06}\,{}^{+0.01}_{-0.01}$ | $\phantom{-}0.03\,^{+0.07}_{-0.07}\,{}^{+0.01}_{-0.01}$ | $\phantom{-}0.07\,^{+0.09}_{-0.08}\,{}^{+0.01}_{-0.01}$
$q^{2}$ $(\mathrm{\,Ge\kern-1.00006ptV}^{2}/c^{4})$ | $A_{9}$ | $A_{\rm T}^{2}$ | $A_{\rm T}^{\rm Re}$ | p-value
$0.10-2.00$ | $\phantom{-}0.12\,_{-0.09}^{+0.09}\,{}^{+0.01}_{-0.01}$ | $-0.14\,_{-0.30}^{+0.34}\,{}^{+0.02}_{-0.02}$ | $-0.04\,_{-0.24}^{+0.26}\,{}^{+0.02}_{-0.01}$ | 0.18
(uncorrected) |  |  |  |
$0.10-2.00$ | $\phantom{-}0.14\,_{-0.11}^{+0.11}\,{}_{-0.01}^{+0.01}$ | $-0.19\,_{-0.35}^{+0.40}\,{}_{-0.02}^{+0.02}$ | $-0.06\,_{-0.27}^{+0.29}\,{}_{-0.01}^{+0.02}$ | –
(corrected) |  |  |  |
$2.00-4.30$ | $\phantom{-}0.06\,_{-0.08}^{+0.12}\,{}^{+0.01}_{-0.01}$ | $-0.29\,_{-0.46}^{+0.65}\,{}^{+0.02}_{-0.03}$ | $-1.00\,_{-0.00}^{+0.13}\,{}^{+0.04}_{-0.00}$ | 0.57
$4.30-8.68$ | $-0.13\,_{-0.07}^{+0.07}\,{}^{+0.01}_{-0.01}$ | $\phantom{-}0.36\,_{-0.31}^{+0.30}\,{}^{+0.03}_{-0.03}$ | $\phantom{-}0.50\,_{-0.14}^{+0.16}\,{}^{+0.01}_{-0.03}$ | 0.71
$10.09-12.86$ | $\phantom{-}0.00\,_{-0.11}^{+0.11}\,{}^{+0.01}_{-0.01}$ | $-0.60\,_{-0.27}^{+0.42}\,{}^{+0.05}_{-0.02}$ | $\phantom{-}0.71\,_{-0.15}^{+0.15}\,{}^{+0.01}_{-0.03}$ | –
$14.18-16.00$ | $-0.06\,_{-0.08}^{+0.11}\,{}^{+0.01}_{-0.01}$ | $\phantom{-}0.07\,_{-0.28}^{+0.26}\,{}^{+0.02}_{-0.02}$ | $\phantom{-}1.00\,_{-0.05}^{+0.00}\,{}^{+0.00}_{-0.02}$ | 0.38
$16.00-19.00$ | $\phantom{-}0.00\,_{-0.10}^{+0.11}\,{}^{+0.01}_{-0.01}$ | $-0.71\,_{-0.26}^{+0.35}\,{}^{+0.06}_{-0.04}$ | $\phantom{-}0.64\,_{-0.15}^{+0.15}\,{}^{+0.01}_{-0.02}$ | 0.28
$1.00-6.00$ | $\phantom{-}0.03\,^{+0.08}_{-0.08}\,{}^{+0.01}_{-0.01}$ | $\phantom{-}0.15\,_{-0.41}^{+0.39}\,{}^{+0.03}_{-0.03}$ | $-0.66\,_{-0.22}^{+0.24}\,{}^{+0.04}_{-0.01}$ | 0.72

Figure 4: Fraction of longitudinal polarisation of the $K^{*0}$, $F_{\rm L}$,
dimuon system forward-backward asymmetry, $A_{\rm FB}$ and the angular
observables $S_{3}$ and $A_{9}$ from the $B^{0}\\!\rightarrow
K^{*0}\mu^{+}\mu^{-}$ decay as a function of the dimuon invariant mass
squared, $q^{2}$. The lowest $q^{2}$ bin has been corrected for the threshold
behaviour described in Sec. 7.2. The experimental data points overlay the SM
prediction described in the text. A rate average of the SM prediction across
each $q^{2}$ bin is indicated by the dark (purple) rectangular regions. No
theory prediction is included for $A_{9}$, which is vanishingly small in the
SM.

Figure 5: Transverse asymmetries $A_{\rm T}^{2}$ and $A_{\rm T}^{\rm Re}$ as a
function of the dimuon invariant mass squared, $q^{2}$, in the
$B^{0}\\!\rightarrow K^{*0}\mu^{+}\mu^{-}$ decay. The lowest $q^{2}$ bin has
been corrected for the threshold behaviour described in Sec. 7.2. The
experimental data points overlay the SM prediction that is described in the
text. A rate average of the SM prediction across each $q^{2}$ bin is indicated
by the dark (purple) rectangular regions.

###  7.2 Angular distribution at large recoil

In the previous section, when fitting the angular distribution, it was assumed
that the muon mass was small compared to that of the dimuon system. Whilst
this assumption is valid for $q^{2}>2\mathrm{\,Ge\kern-1.00006ptV}^{2}/c^{4}$,
it breaks down in the $0.1<q^{2}<2.0\mathrm{\,Ge\kern-1.00006ptV}^{2}/c^{4}$
bin. In this bin, the angular terms receive an additional $q^{2}$ dependence,
proportional to

$\frac{1-{4m_{\mu}^{2}}/{q^{2}}}{1+2m_{\mu}^{2}/q^{2}}~{}~{}\text{or}~{}~{}\frac{(1-4m_{\mu}^{2}/q^{2})^{1/2}}{1+2m_{\mu}^{2}/q^{2}}~{},$
(6)

depending on the angular term $I_{j}$ [1].

As $q^{2}$ tends to zero, these threshold terms become small and reduce the
sensitivity to the angular observables. Neglecting these terms leads to a bias
in the measurement of the angular observables. Previous analyses by LHCb,
BaBar, Belle and CDF have not considered this effect.

The fraction of longitudinal polarisation of the $K^{*0}$ meson, $F_{\rm L}$,
is the only observable that is unaffected by the additional terms; sensitivity
to $F_{\rm L}$ arises mainly through the shape of the $\cos\theta_{K}$
distribution and this shape remains the same whether the threshold terms are
included or not.

In order to estimate the size of the bias, it is assumed that $A_{9}$ and
$A_{T}^{2}$ are constant over the
$0.1<q^{2}<2\mathrm{\,Ge\kern-1.00006ptV}^{2}/c^{4}$ region and $A_{\rm
T}^{\rm Re}$ rises linearly (with the constraint that $A_{\rm T}^{\rm Re}=0$
at $q^{2}=0$). Even though $F_{\rm L}$ is in itself unbiased, an assumption
needs to be made about the $q^{2}$ dependence of $F_{\rm L}$ when determining
the bias introduced on the other observables. An empirical model,

$F_{\rm L}(q^{2})=\frac{aq^{2}}{1+aq^{2}}~{}~{},$ (7)

is used. This functional form displays the correct behaviour since it tends to
zero as $q^{2}$ tends to zero and rises slowly over the $q^{2}$ bin,
reflecting the dominance of the photon penguin at low $q^{2}$ and the
transverse polarisation of the photon.

The coefficient $a=0.67\,^{+0.54}_{-0.30}$ is estimated by assigning each
(background subtracted) signal candidate a value of $F_{\rm L}$ according to
Eq. 7, averaging $F_{\rm L}$ over the candidates in the $q^{2}$ bin and
comparing this to the value that is obtained from the fit to the
$0.1<q^{2}<2.0\mathrm{\,Ge\kern-1.00006ptV}^{2}/c^{4}$ region (in Table 2).
Different values of the coefficient $a$ are tried until the two estimates
agree.

To remain model independent, the bias on the angular observables is similarly
estimated by summing over the observed candidates. A concrete example of how
this is done is given in Appendix B for the observable $A_{\rm T}^{2}$. The
typical size of the correction is $10-20\%$. The values of the angular
observables, after correcting for the bias, are included in Table 2. A similar
factor is also applied to the statistical uncertainty on the fit parameters to
scale them accordingly. No systematic uncertainty is assigned to this
correction.

The procedure to calculate the size of the bias that is introduced by
neglecting the threshold terms has been validated using large samples of
simulated events, generated according to the SM prediction and several other
scenarios in which large deviations from the SM expectation of the angular
observables are possible. In all cases an unbiased estimate of the angular
observables is obtained after applying the correction procedure. Different
hypotheses for the $q^{2}$ dependence of $F_{\rm L}$, $A_{\rm FB}$ and $A_{\rm
T}^{\rm Re}$ do not give large variations in the size of the correction
factors.

###  7.3 Systematic uncertainties in the angular analysis

Sources of systematic uncertainty are considered if they introduce either an
angular or $q^{2}$ dependent bias to the acceptance correction. Moreover,
three assumptions have been made that may affect the interpretation of the
result of the fit to the $K^{+}\pi^{-}\mu^{+}\mu^{-}$ invariant mass or
angular distribution: that $q^{2}\gg 4m_{\mu}^{2}$; that there are equal
numbers of $B^{0}$ and $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}$
decays; and that there is no contribution from non-$K^{*0}$
$B^{0}\\!\rightarrow K^{+}\pi^{-}\mu^{+}\mu^{-}$ decays in the
$792<m({K^{+}\pi^{-}})<992{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ mass
window. The first assumption was addressed in Sec. 7.2 and no systematic
uncertainty is assigned to this correction. The number of $B^{0}$ and $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ candidates in the data set is very
similar [38]. The resulting systematic uncertainty is addressed in Sec. 7.3.1.
The final assumption is discussed in Sec. 7.3.2 below.

The full fitting procedure has been tested on $B^{0}\\!\rightarrow
K^{*0}{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ decays. In this larger
data sample, $A_{\rm FB}$ is found to be consistent with zero (as expected)
and the other observables are in agreement with the results of Ref. [39].
There is however a small discrepancy between the expected parabolic shape of
the $\cos\theta_{K}$ distribution and the distribution of the
$B^{0}\\!\rightarrow K^{*0}{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$
candidates after weighting the candidates to correct for the detector
acceptance. This percent-level discrepancy could point to a bias in the
acceptance model. To account for this discrepancy, and any breakdown in the
assumption that the efficiencies in $\cos\theta_{\ell}$, $\cos\theta_{K}$ and
$\phi$ are independent, systematic variations of the weights are tried in
which they are conservatively rescaled by 10% at the edges of
$\cos\theta_{\ell}$, $\cos\theta_{K}$ and $\phi$ with respect to the centre.
Several possible variations are explored, including variations that are non-
factorisable. The variation which has the largest effect on each of the
angular observables is assigned as a systematic uncertainty. The resulting
systematic uncertainties are at the level of $0.01-0.03$ and are largest for
the transverse observables.

The uncertainties on the signal mass model have little effect on the angular
observables. Of more importance are potential sources of uncertainty on the
background shape. In the angular fit the background is modelled as the product
of three second-order polynomials, the parameters of which are allowed to vary
freely in the likelihood fit. This model describes the data well in the
sidebands. As a cross-check, alternative fits are performed both using higher
order polynomials and by fixing the shape of the background to be flat in
$\cos\theta_{\ell}$, $\cos\theta_{K}$ and $\hat{\phi}$. The largest shifts in
the angular observables occur for the flat background model and are at the
level of $0.01-0.06$ and $0.02-0.25$ for the transverse observables (they are
at most 65% of the statistical uncertainty). These variations are extreme
modifications of the background model and are not considered further as
sources of systematic uncertainty.

The angular distributions of the decays
$B^{0}_{s}\\!\rightarrow\phi\mu^{+}\mu^{-}$ and $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\\!\rightarrow
K^{*0}\mu^{+}\mu^{-}$ are both poorly known. The decay $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\\!\rightarrow
K^{*0}\mu^{+}\mu^{-}$ is yet to be observed. A first measurement of
$B^{0}_{s}\\!\rightarrow\phi\mu^{+}\mu^{-}$ has been made in Ref. [40]. In the
likelihood fit to the angular distribution these backgrounds are neglected. A
conservative systematic uncertainty on the angular observables is assigned at
the level of ${~{}\raise 1.49994pt\hbox{$<$}\kern-8.50006pt\lower
3.50006pt\hbox{$\sim$}~{}}0.01$ by assuming that the peaking backgrounds have
an identical shape to the signal, but have an angular distribution in which
each of the observables is either maximal or minimal.

Systematic variations are also considered for the data-derived corrections to
the simulated events. For example, the muon identification efficiency, which
is derived from data using a tag-and-probe approach with
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ decays, is varied within its
uncertainty in opposite direction for high
($p>10{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$) and low
($p<10{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$) momentum muons. Similar
variations are applied to the other data-derived corrections, yielding a
combined systematic uncertainty at the level of $0.01-0.02$ on the angular
observables. The correction needed to account for differences between data and
simulation in the $B^{0}$ momentum spectrum is small. If this correction is
neglected, the angular observables vary by at most 0.01. This variation is
associated as a systematic uncertainty.

The systematic uncertainties arising from the variations of the angular
acceptance are assessed using pseudo-experiments that are generated with one
acceptance model and fitted according to a different model. Consistent results
are achieved by varying the event weights applied to the data and repeating
the likelihood fit.

A summary of the different contributions to the total systematic uncertainty
can be found in Table 3. The systematic uncertainty on the angular observables
in Table 2 is the result of adding these contributions in quadrature.

Table 3:  Systematic contributions to the angular observables. The values given are the magnitude of the maximum contribution from each source of systematic uncertainty, taken across the six principal $q^{2}$ bins used in the analysis. Source | $A_{\rm FB}$ | $F_{\rm L}$ | $S_{3}$ | $S_{9}$ | $A_{9}$ | $A_{\rm T}^{2}$ | $A_{\rm T}^{\rm Re}$
---|---|---|---|---|---|---|---
Acceptance model | $\phantom{<}0.02$ | $\phantom{<}0.03$ | $\phantom{<}0.01$ | $<0.01$ | $<0.01$ | $\phantom{<}0.02$ | $\phantom{<}0.01$
Mass model | $<0.01$ | $<0.01$ | $<0.01$ | $<0.01$ | $<0.01$ | $<0.01$ | $<0.01$
$B^{0}\rightarrow\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ mis-id | $<0.01$ | $<0.01$ | $<0.01$ | $<0.01$ | $\phantom{<}0.01$ | $<0.01$ | $<0.01$
Data-simulation diff. | $\phantom{<}0.01$ | $\phantom{<}0.03$ | $\phantom{<}0.01$ | $<0.01$ | $<0.01$ | $\phantom{<}0.03$ | $\phantom{<}0.01$
Kinematic reweighting | $<0.01$ | $\phantom{<}0.01$ | $<0.01$ | $<0.01$ | $<0.01$ | $\phantom{<}0.01$ | $<0.01$
Peaking backgrounds | $\phantom{<}0.01$ | $\phantom{<}0.01$ | $\phantom{<}0.01$ | $\phantom{<}0.01$ | $\phantom{<}0.01$ | $\phantom{<}0.01$ | $\phantom{<}0.01$
S-wave | $\phantom{<}0.01$ | $\phantom{<}0.01$ | $\phantom{<}0.02$ | $\phantom{<}0.01$ | $<0.01$ | $\phantom{<}0.05$ | $\phantom{<}0.04$
$B^{0}$-$\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ asymmetries | $<0.01$ | $<0.01$ | $<0.01$ | $<0.01$ | $<0.01$ | $<0.01$ | $<0.01$

####  7.3.1 Production, detection and direct ${\boldmath C\\!P}$ asymmetries

If the number of $B^{0}$ and $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ decays are not equal in the
likelihood fit then the terms in the angular distribution no longer correspond
to pure $C\\!P$ averages or asymmetries. They instead correspond to admixtures
of the two, e.g.

$S_{3}^{\text{obs}}\approx S_{3}-A_{3}\left(\mathcal{A}_{\rm
CP}+\kappa\mathcal{A}_{\rm P}+\mathcal{A}_{\rm D}\right)~{},$ (8)

where $\mathcal{A}_{\rm CP}$ is the direct $C\\!P$ asymmetry between
$B^{0}\\!\rightarrow K^{*0}\mu^{+}\mu^{-}$ and $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\\!\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}\mu^{+}\mu^{-}$ decays;
$\mathcal{A}_{\rm P}$ is the production asymmetry between $B^{0}$ and $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ mesons, which is diluted by a
factor $\kappa$ due to $B^{0}-\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ mixing; and $\mathcal{A}_{\rm D}$
is the detection asymmetry between the $B^{0}$ and $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ decays (which might be non-zero
due to differences in the interaction cross-section with matter between
$K^{+}$ and $K^{-}$ mesons). In practice, the production and detection
asymmetries are small in LHCb and $\mathcal{A}_{\rm CP}$ is measured to be
$\mathcal{A}_{\rm CP}=-0.072\pm 0.040\pm 0.005$ [38], which is consistent with
zero. Combined with the expected small size of the $C\\!P$ asymmetry or
$C\\!P$-averaged counterparts of the angular observables measured in this
analysis, this reduces any systematic bias to $<0.01$.

####  7.3.2 Influence of S-wave interference on the angular distribution

The presence of a non-$K^{*0}$ $B^{0}\\!\rightarrow
K^{+}\pi^{-}\mu^{+}\mu^{-}$ component, where the $K^{+}\pi^{-}$ system is in
an S-wave configuration, modifies Eq. 4 to

$\begin{split}\frac{1}{\mathrm{d}\Gamma^{\prime}/\mathrm{d}q^{2}}\frac{\mathrm{d}^{4}\Gamma^{\prime}}{\mathrm{d}q^{2}\,\mathrm{d}\cos\theta_{\ell}\,\mathrm{d}\cos\theta_{K}\,\mathrm{d}\hat{\phi}}=&~{}(1-F_{\rm
S})\left[\frac{1}{\mathrm{d}\Gamma/\mathrm{d}q^{2}}\frac{\mathrm{d}^{4}\Gamma}{\mathrm{d}q^{2}\,\mathrm{d}\cos\theta_{\ell}\,\mathrm{d}\cos\theta_{K}\,\mathrm{d}\hat{\phi}}\right]\\\
&~{}~{}\,+\frac{9}{16\pi}\left[\frac{2}{3}F_{\rm
S}(1-\cos^{2}\theta_{\ell})+\frac{4}{3}A_{\rm
S}\cos\theta_{K}(1-\cos^{2}\theta_{\ell})\right]~{},\\\ \end{split}$ (9)

where $F_{\rm S}$ is the fraction of $B^{0}\\!\rightarrow
K^{+}\pi^{-}\mu^{+}\mu^{-}$ S-wave in the
$792<m({K^{+}\pi^{-}})<992{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ window.
The partial width, $\Gamma^{\prime}$, is the sum of the partial widths for the
$B^{0}\\!\rightarrow K^{*0}\mu^{+}\mu^{-}$ decay and the $B^{0}\\!\rightarrow
K^{+}\pi^{-}\mu^{+}\mu^{-}$ S-wave. A forward-backward asymmetry in
$\cos\theta_{K}$, $A_{\rm S}$, arises due to the interference between the
longitudinal amplitude of the $K^{*0}$ and the S-wave amplitude [41, 42, 43,
44].

The S-wave is neglected in the results given in Table 2. To estimate the size
of the S-wave component, and the impact it might have on the
$B^{0}\\!\rightarrow K^{*0}\mu^{+}\mu^{-}$ angular analysis, the phase shift
of the $K^{*0}$ Breit-Wigner function around the $K^{*0}$ pole mass is
exploited. Instead of measuring $F_{\rm S}$ directly, the average value of
$A_{\rm S}$ is measured in two bins of $K^{+}\pi^{-}$ invariant mass, one
below and one above the $K^{*0}$ pole mass. If the magnitude and phase of the
S-wave amplitude are assumed to be independent of the $K^{+}\pi^{-}$ invariant
mass in the range
$792<m({K^{+}\pi^{-}})<992{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$, and the
P-wave amplitude is modelled by a Breit-Wigner function, the two $A_{\rm S}$
values can then be used to determine the real and imaginary components of the
S-wave amplitude (and $F_{\rm S}$).444In the decay $B^{0}\\!\rightarrow
K^{*0}\mu^{+}\mu^{-}$ there are actually two pairs of amplitudes involved,
left- and right-handed longitudinal amplitudes and left- and right-handed
S-wave amplitudes (where the handedness refers to the chirality of the dimuon
system). In order to exploit the interference and determine $F_{\rm S}$ it is
assumed that the phase difference between the two left-handed amplitudes is
the same as the difference between the two right-handed amplitudes, as
expected from the expression for the amplitudes in Refs. [41, 42].

For a small S-wave amplitude, the pure S-wave contribution, $F_{\rm S}$, to
Eq. 9 has only a small effect on the angular distribution. The magnitude of
$A_{\rm S}$ arising from the interference between the S- and P-wave can
however still be sizable and this information is exploited by this phase-shift
method. The method, described above, is statistically more precise than
fitting Eq. 9 directly for $A_{\rm S}$ and $F_{\rm S}$ as uncorrelated
variables. For the $B^{0}\\!\rightarrow
K^{*0}{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ control mode, the gain in
statistical precision is approximately a factor of three.

Due to the limited number of signal candidates that are available in each of
the $q^{2}$ bins, the bins are merged in order to estimate the S-wave
fraction. In the range $0.1<q^{2}<19\mathrm{\,Ge\kern-1.00006ptV}^{2}/c^{4}$,
$F_{\rm S}=0.03\pm 0.03$, which corresponds to an upper limit of $F_{\rm
S}<0.04$ at $68\%$ confidence level (CL). The procedure has also been
performed in the region $1<q^{2}<6\mathrm{\,Ge\kern-1.00006ptV}^{2}/c^{4}$,
where both $F_{\rm L}$ and $F_{\rm S}$ are expected to be enhanced. This gives
$F_{\rm S}=0.04\pm 0.04$ and an upper limit of $F_{\rm S}<0.07$ at $68\%$ CL.
In order to be conservative, $F_{\rm S}=0.07$ is used to estimate a systematic
uncertainty on the differential branching fraction and angular analyses. The
$B^{0}\\!\rightarrow K^{*0}{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ data
has been used to validate the method.

For the differential branching fraction analysis, $F_{\rm S}$ scales the
observed branching fraction by up to 7%. For the angular analysis, $F_{\rm S}$
dilutes $A_{\rm FB}$, $S_{3}$ and $A_{9}$. The impact on $F_{\rm L}$ however,
is less easy to disentangle. To assess the possible size of a systematic bias,
pseudo-experiments have been carried out generating with, and fitting without,
the S-wave contribution in the likelihood fit. The typical bias on the angular
observables due to the S-wave is $0.01-0.03$.

##  8 Forward-backward asymmetry zero-crossing point

In the SM, $A_{\rm FB}$ changes sign at a well defined value of $q^{2}$,
$q^{2}_{0}$, whose prediction is largely free from form-factor uncertainties
[3]. It is non-trivial to estimate $q^{2}_{0}$ from the angular fits to the
data in the different $q^{2}$ bins, due to the large size of the bins
involved. Instead, $A_{\rm FB}$ can be estimated by counting the number of
forward-going ($\cos\theta_{\ell}>0$) and backward-going
($\cos\theta_{\ell}<0$) candidates and $q^{2}_{0}$ determined from the
resulting distribution of $A_{\rm FB}(q^{2})$.

The $q^{2}$ distribution of the forward- and backward-going candidates, in the
range $1.0<q^{2}<7.8\mathrm{\,Ge\kern-1.00006ptV}^{2}/c^{4}$, is shown in Fig.
6. To make a precise measurement of the zero-crossing point a polynomial fit,
$P(q^{2})$, is made to the $q^{2}$ distributions of these candidates. The
$K^{+}\pi^{-}\mu^{+}\mu^{-}$ invariant mass is included in the fit to separate
signal from background. If $P_{\rm F}(q^{2})$ describes the $q^{2}$ dependence
of the forward-going, and $P_{\rm B}(q^{2})$ the backward-going signal decays,
then

$A_{\rm FB}(q^{2})=\frac{P_{\rm F}(q^{2})-P_{\rm B}(q^{2})}{P_{\rm
F}(q^{2})+P_{\rm B}(q^{2})}~{}~{}.$ (10)

The zero-crossing point of $A_{\rm FB}$ is found by solving for the value of
$q^{2}$ at which $A_{\rm FB}(q^{2})$ is zero.

Using third-order polynomials to describe both the $q^{2}$ dependence of the
signal and the background, the zero-crossing point is found to be

$q_{0}^{2}=4.9\pm 0.9\mathrm{\,Ge\kern-1.00006ptV}^{2}/c^{4}~{}~{}.$

The uncertainty on $q^{2}_{0}$ is determined using a bootstrapping technique
[45]. The zero-crossing point is largely independent of the polynomial order
and the $q^{2}$ range that is used. This value is consistent with SM
predictions, which are typically in the range
$3.9-4.4\mathrm{\,Ge\kern-1.00006ptV}^{2}/c^{4}$ [46, 47, 48] and have
relative uncertainties below the 10% level, for example,
$q_{0}^{2}=4.36\,^{+0.33}_{-0.31}\mathrm{\,Ge\kern-1.00006ptV}^{2}/c^{4}$
[47].

The systematic uncertainty on the zero-crossing point of the forward-backward
asymmetry is negligible compared to the statistical uncertainty. To generate a
large systematic bias, it would be necessary to create an asymmetric
acceptance effect in $\cos\theta_{\ell}$ that is not canceled when combining
$B^{0}$ and $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ decays. The
combined systematic uncertainty is at the level of $\pm
0.05\mathrm{\,Ge\kern-1.00006ptV}^{2}/c^{4}$.

Figure 6:  Dimuon invariant mass squared, $q^{2}$, distribution of forward-
going (left) and backward-going (right) candidates in the
$K^{+}\pi^{-}\mu^{+}\mu^{-}$ invariant mass window
$5230<m(K^{+}\pi^{-}\mu^{+}\mu^{-})<5330{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$.
The polynomial fit to the signal and background distributions in $q^{2}$ is
overlaid.

##  9 Conclusions

In summary, using a data sample corresponding to 1.0$\mbox{\,fb}^{-1}$ of
integrated luminosity, collected by the LHCb experiment in 2011, the
differential branching fraction, $\mathrm{d}{\cal B}/\mathrm{d}q^{2}$, of the
decay $B^{0}\\!\rightarrow K^{*0}\mu^{+}\mu^{-}$ has been measured in bins of
$q^{2}$. Measurements of the angular observables, $A_{\rm FB}$ ($A_{\rm
T}^{\rm Re}$), $F_{L}$, $S_{3}$ ($A_{\rm T}^{2}$) and $A_{9}$ have also been
performed in the same $q^{2}$ bins.

The complete set of results obtained in this paper are provided in Tables 1
and 2. These are the most precise measurements of $\mathrm{d}$$\cal
B$/$\mathrm{d}$$q^{2}$ and the angular observables to date. All of the
observables are consistent with SM expectations and together put stringent
constraints on the contributions from new particles to $b\rightarrow s$
flavour changing neutral current processes. A bin-by-bin comparison of the
reduced angular distribution with the SM hypothesis indicates an excellent
agreement with p-values between 18 and 72%.

Finally, a first measurement of the zero-crossing point of the forward-
backward asymmetry has also been performed, yielding $q_{0}^{2}=4.9\pm
0.9\mathrm{\,Ge\kern-1.00006ptV}^{2}/c^{4}$. This measurement is again
consistent with SM expectations.

## 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); MEN/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.

Appendix

##  Appendix A Angular basis

Figure 7:  Graphical representation of the angular basis used for
$B^{0}\\!\rightarrow K^{*0}\mu^{+}\mu^{-}$ and $\kern
1.61993pt\overline{\kern-1.61993ptB}{}^{0}\\!\rightarrow\kern
1.79997pt\overline{\kern-1.79997ptK}{}^{*0}\mu^{+}\mu^{-}$ decays in this
paper. The notation $\hat{n}_{ab}$ is used to represent the normal to the
plane containing particles $a$ and $b$ in the $B^{0}$ (or $\kern
1.61993pt\overline{\kern-1.61993ptB}{}^{0}$) rest frame. An explicit
description of the angular basis is given in the text.

The angular basis used in this paper is illustrated in Fig. 7. The angle
$\theta_{\ell}$ is defined as the angle between the direction of the $\mu^{+}$
($\mu^{-}$) in the dimuon rest frame and the direction of the dimuon in the
$B^{0}$ ($\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}$) rest frame. The
angle $\theta_{K}$ is defined as the angle between the direction of the kaon
in the $K^{*0}$ ($\kern 1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$) rest
frame and the direction of the $K^{*0}$ ($\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$) in the $B^{0}$ ($\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}$) rest frame. The angle $\phi$ is
the angle between the plane containing the $\mu^{+}$ and $\mu^{-}$ and the
plane containing the kaon and pion from the $K^{*0}$. Explicitly,
$\cos\theta_{\ell}$ and $\cos\theta_{K}$ are defined as

$\cos\theta_{\ell}=\left(\hat{p}_{\mu^{+}}^{(\mu^{+}\mu^{-})}\right)\cdot\left(\hat{p}_{\mu^{+}\mu^{-}}^{(B^{0})}\right)=\left(\hat{p}_{\mu^{+}}^{(\mu^{+}\mu^{-})}\right)\cdot\left(-\hat{p}_{B^{0}}^{(\mu^{+}\mu^{-})}\right)~{},$
(11)

$\cos\theta_{K}=\left(\hat{p}_{K^{+}}^{(K^{*0})}\right)\cdot\left(\hat{p}_{K^{*0}}^{(B^{0})}\right)=\left(\hat{p}_{K^{+}}^{(K^{*0})}\right)\cdot\left(-\hat{p}_{B^{0}}^{(K^{*0})}\right)$
(12)

for the $B^{0}$ and

$\cos\theta_{\ell}=\left(\hat{p}_{\mu^{-}}^{(\mu^{+}\mu^{-})}\right)\cdot\left(\hat{p}_{\mu^{+}\mu^{-}}^{(\kern
1.25995pt\overline{\kern-1.25995ptB}{}^{0})}\right)=\left(\hat{p}_{\mu^{-}}^{(\mu^{+}\mu^{-})}\right)\cdot\left(-\hat{p}_{\kern
1.25995pt\overline{\kern-1.25995ptB}{}^{0}}^{(\mu^{+}\mu^{-})}\right)~{},$
(13)

$\cos\theta_{K}=\left(\hat{p}_{K^{-}}^{(K^{*0})}\right)\cdot\left(\hat{p}_{K^{*0}}^{(\kern
1.25995pt\overline{\kern-1.25995ptB}{}^{0})}\right)=\left(\hat{p}_{K^{-}}^{(K^{*0})}\right)\cdot\left(-\hat{p}_{\kern
1.25995pt\overline{\kern-1.25995ptB}{}^{0}}^{(K^{*0})}\right)$ (14)

for the $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ decay. The
definition of the angle $\phi$ is given by

$\cos\phi=\left(\hat{p}_{\mu^{+}}^{(B^{0})}\times\hat{p}_{\mu^{-}}^{(B^{0})}\right)\cdot\left(\hat{p}_{K^{+}}^{(B^{0})}\times\hat{p}_{\pi^{-}}^{(B^{0})}\right)~{},$
(15)

$\sin\phi=\left[\left(\hat{p}_{\mu^{+}}^{(B^{0})}\times\hat{p}_{\mu^{-}}^{(B^{0})}\right)\times\left(\hat{p}_{K^{+}}^{(B^{0})}\times\hat{p}_{\pi^{-}}^{(B^{0})}\right)\right]\cdot\hat{p}_{K^{*0}}^{(B^{0})}$
(16)

for the $B^{0}$ and

$\cos\phi=\left(\hat{p}_{\mu^{-}}^{(\kern
1.25995pt\overline{\kern-1.25995ptB}{}^{0})}\times\hat{p}_{\mu^{+}}^{(\kern
1.25995pt\overline{\kern-1.25995ptB}{}^{0})}\right)\cdot\left(\hat{p}_{K^{-}}^{(\kern
1.25995pt\overline{\kern-1.25995ptB}{}^{0})}\times\hat{p}_{\pi^{+}}^{(\kern
1.25995pt\overline{\kern-1.25995ptB}{}^{0})}\right)~{},$ (17)

$\sin\phi=-\left[\left(\hat{p}_{\mu^{-}}^{(\kern
1.25995pt\overline{\kern-1.25995ptB}{}^{0})}\times\hat{p}_{\mu^{+}}^{(\kern
1.25995pt\overline{\kern-1.25995ptB}{}^{0})}\right)\times\left(\hat{p}_{K^{-}}^{(\kern
1.25995pt\overline{\kern-1.25995ptB}{}^{0})}\times\hat{p}_{\pi^{+}}^{(\kern
1.25995pt\overline{\kern-1.25995ptB}{}^{0})}\right)\right]\cdot\hat{p}_{\kern
1.39998pt\overline{\kern-1.39998ptK}{}^{*0}}^{(\kern
1.25995pt\overline{\kern-1.25995ptB}{}^{0})}$ (18)

for the $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ decay. The
$\hat{p}_{X}^{(Y)}$ are unit vectors describing the direction of a particle
$X$ in the rest frame of the system $Y$. In every case the particle momenta
are first boosted to the $B^{0}$ (or $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}$) rest frame. In this basis, the
angular definition for the $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}$
decay is a $C\\!P$ transformation of that for the $B^{0}$ decay.

##  Appendix B Angular distribution at large recoil

An explicit example of the bias on the angular observables that comes from the
threshold terms is provided below for $A_{\rm T}^{2}$. Sensitivity to $A_{\rm
T}^{2}$ comes through the term in Eq. 1 with
$\sin^{2}\theta_{\ell}\sin^{2}\theta_{K}\cos 2\phi$ angular dependence. In the
limit $q^{2}\gg m_{\mu}^{2}$, this term is simply

$\frac{1}{2}\left(1-F_{\rm L}(q^{2})\right)A_{\rm
T}^{2}(q^{2})\sin^{2}\theta_{\ell}\sin^{2}\theta_{K}\cos 2\phi$ (19)

and the differential decay width is

$\frac{\mathrm{d}\Gamma}{\mathrm{d}q^{2}}=|A_{0,{\rm
L}}|^{2}+|A_{\parallel,{\rm L}}|^{2}+|A_{\perp,{\rm L}}|^{2}+|A_{0,{\rm
R}}|^{2}+|A_{\parallel,{\rm R}}|^{2}+|A_{\perp,{\rm R}}|^{2}~{},$ (20)

where $A_{0}$, $A_{\parallel}$ and $A_{\perp}$ are the $K^{*0}$ spin-
amplitudes and the L/R index refers to the chirality of the lepton current
(see for example Ref. [1]). If $q^{2}{~{}\raise
1.49994pt\hbox{$<$}\kern-8.50006pt\lower
3.50006pt\hbox{$\sim$}~{}}1\mathrm{\,Ge\kern-1.00006ptV}^{2}/c^{4}$, these
expressions are modified to

$\frac{1}{2}\left[\frac{1-4m_{\mu}^{2}/q^{2}}{1+2m_{\mu}^{2}/q^{2}}\right]\left(1-F_{\rm
L}(q^{2})\right)A_{\rm
T}^{2}(q^{2})\sin^{2}\theta_{\ell}\sin^{2}\theta_{K}\cos 2\phi$ (21)

and

$\frac{\mathrm{d}\Gamma}{\mathrm{d}q^{2}}=\left[1+2m_{\mu}^{2}/q^{2}\right]\left(|A_{0,{\rm
L}}|^{2}+|A_{\parallel,{\rm L}}|^{2}+|A_{\perp,{\rm L}}|^{2}+|A_{0,{\rm
R}}|^{2}+|A_{\parallel,{\rm R}}|^{2}+|A_{\perp,{\rm R}}|^{2}\right).$ (22)

In an infinitesimal window of $q^{2}$, the difference between an experimental
measurement of $A_{\rm T}^{2}$, $A_{\rm T}^{2~{}\text{exp}}$, in which the
threshold terms are neglected and the value of $A_{\rm T}^{2}$ defined in
literature is

$\frac{A_{\rm T}^{2~{}\text{exp}}}{A_{\rm
T}^{2}}=\left[\frac{1-4m_{\mu}^{2}/q^{2}}{1+2m_{\mu}^{2}/q^{2}}\right]~{}~{}.$
(23)

Unfortunately, in a wider $q^{2}$ window, the $q^{2}$ dependence of $F_{\rm
L}$, $A_{\rm T}^{2}$ and the threshold terms needs to be considered and it
becomes less straightforward to estimate the bias due to the threshold terms.
If $A_{\rm T}^{2}$ is constant over the $q^{2}$ window,

$\frac{A_{\rm T}^{2~{}\text{exp}}}{A_{\rm
T}^{2}}=\frac{\displaystyle\int_{q^{2}_{\text{min}}}^{q^{2}_{\text{max}}}\frac{\mathrm{d}\Gamma}{\mathrm{d}q^{2}}\left[\frac{1-4m_{\mu}^{2}/q^{2}}{1+2m_{\mu}^{2}/q^{2}}\right]\left[1-F_{\rm
L}(q^{2})\right]\mathrm{d}q^{2}}{\displaystyle\int_{q^{2}_{\text{min}}}^{q^{2}_{\text{max}}}\frac{\mathrm{d}\Gamma}{\mathrm{d}q^{2}}\left[1-F_{\rm
L}(q^{2})\right]\mathrm{d}q^{2}}~{}~{}.$ (24)

In practice the integration in Eq. 24 can be replaced by a sum over the signal
events in the $q^{2}$ window

$\frac{A_{\rm T}^{2~{}\text{exp}}}{A_{\rm
T}^{2}}=\frac{\sum\limits_{i=0}^{N}\left[\frac{1-4m_{\mu}^{2}/q^{2}_{i}}{1+2m_{\mu}^{2}/q^{2}_{i}}\right](1-F_{\rm
L}(q_{i}^{2}))\omega_{i}}{\sum\limits_{i=0}^{N}(1-F_{\rm
L}(q_{i}^{2}))\omega_{i}}~{},$ (25)

where $\omega_{i}$ is a weight applied to the $i^{\text{th}}$ candidate to
account for the detector and selection acceptance and the background in the
$q^{2}$ window.

Correction factors for the other observables can be similarly defined if it is
assumed that they are constant over the $q^{2}$ window. In the case of $A_{\rm
FB}$ (and $A_{\rm T}^{\rm Re}$) that are expected to exhibit a strong $q^{2}$
dependence, the $q^{2}$ dependence of the observable needs to be considered.

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