EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)

 CERN-PH-EP-2014-038 LHCb-PAPER-2014-005

Measurement of polarization amplitudes and $C\\!P$ asymmetries in
$B^{0}\rightarrow\phi K^{*}(892)^{0}$

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

An angular analysis of the decay $B^{0}\rightarrow\phi K^{*}(892)^{0}$ is
reported based on a $pp$ collision data sample, corresponding to an integrated
luminosity of $1.0\mbox{\,fb}^{-1}$, collected at a centre-of-mass energy of
$\sqrt{s}=7\,\mathrm{\,Te\kern-1.00006ptV}$ with the LHCb detector. The P-wave
amplitudes and phases are measured with a greater precision than by previous
experiments, and confirm about equal amounts of longitudinal and transverse
polarization. The S-wave $K^{+}\pi^{-}$ and $K^{+}K^{-}$ contributions are
taken into account and found to be significant. A comparison of the
$B^{0}\rightarrow\phi K^{*}(892)^{0}$ and $\kern
1.99997pt\overline{\kern-1.99997ptB}{}^{0}\rightarrow\phi\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*}(892)^{0}$ results shows no evidence
for direct $C\\!P$ violation in the rate asymmetry, in the triple-product
asymmetries or in the polarization amplitudes and phases.

Submitted to the Journal of High Energy Physics

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

LHCb collaboration

R. Aaij41, A. Abba21,u, B. Adeva37, M. Adinolfi46, A. Affolder52, Z.
Ajaltouni5, J. Albrecht9, F. Alessio38, M. Alexander51, S. Ali41, G.
Alkhazov30, P. Alvarez Cartelle37, A.A. Alves Jr25,38, S. Amato2, S. Amerio22,
Y. Amhis7, L. An3, L. Anderlini17,g, J. Anderson40, R. Andreassen57, M.
Andreotti16,f, J.E. Andrews58, R.B. Appleby54, O. Aquines Gutierrez10, F.
Archilli38, A. Artamonov35, M. Artuso59, E. Aslanides6, G. Auriemma25,n, M.
Baalouch5, S. Bachmann11, J.J. Back48, A. Badalov36, V. Balagura31, W.
Baldini16, R.J. Barlow54, C. Barschel38, S. Barsuk7, W. Barter47, V.
Batozskaya28, Th. Bauer41, A. Bay39, J. Beddow51, F. Bedeschi23, I. Bediaga1,
S. Belogurov31, K. Belous35, I. Belyaev31, E. Ben-Haim8, G. Bencivenni18, S.
Benson50, J. Benton46, A. Berezhnoy32, R. Bernet40, M.-O. Bettler47, M. van
Beuzekom41, A. Bien11, S. Bifani45, T. Bird54, A. Bizzeti17,i, P.M.
Bjørnstad54, T. Blake48, F. Blanc39, J. Blouw10, S. Blusk59, V. Bocci25, A.
Bondar34, N. Bondar30,38, W. Bonivento15,38, S. Borghi54, A. Borgia59, M.
Borsato7, T.J.V. Bowcock52, E. Bowen40, C. Bozzi16, T. Brambach9, J. van den
Brand42, J. Bressieux39, D. Brett54, M. Britsch10, T. Britton59, N.H. Brook46,
H. Brown52, A. Bursche40, G. Busetto22,q, J. Buytaert38, S. Cadeddu15, R.
Calabrese16,f, O. Callot7, M. Calvi20,k, M. Calvo Gomez36,o, A. Camboni36, P.
Campana18,38, D. Campora Perez38, F. Caponio21,u, A. Carbone14,d, G.
Carboni24,l, R. Cardinale19,38,j, A. Cardini15, H. Carranza-Mejia50, L.
Carson50, K. Carvalho Akiba2, G. Casse52, L. Cassina20, L. Castillo Garcia38,
M. Cattaneo38, Ch. Cauet9, R. Cenci58, M. Charles8, Ph. Charpentier38, S.-F.
Cheung55, N. Chiapolini40, M. Chrzaszcz40,26, K. Ciba38, X. Cid Vidal38, G.
Ciezarek53, P.E.L. Clarke50, M. Clemencic38, H.V. Cliff47, J. Closier38, C.
Coca29, V. Coco38, J. Cogan6, E. Cogneras5, P. Collins38, A. Comerma-
Montells36, A. Contu15,38, A. Cook46, M. Coombes46, S. Coquereau8, G. Corti38,
M. Corvo16,f, I. Counts56, B. Couturier38, G.A. Cowan50, D.C. Craik48, M. Cruz
Torres60, S. Cunliffe53, R. Currie50, C. D’Ambrosio38, J. Dalseno46, P.
David8, P.N.Y. David41, A. Davis57, K. De Bruyn41, S. De Capua54, M. De
Cian11, J.M. De Miranda1, L. De Paula2, W. De Silva57, P. De Simone18, D.
Decamp4, M. Deckenhoff9, L. Del Buono8, N. Déléage4, D. Derkach55, O.
Deschamps5, F. Dettori42, A. Di Canto38, H. Dijkstra38, S. Donleavy52, F.
Dordei11, M. Dorigo39, A. Dosil Suárez37, D. Dossett48, A. Dovbnya43, F.
Dupertuis39, P. Durante38, R. Dzhelyadin35, A. Dziurda26, A. Dzyuba30, S.
Easo49, U. Egede53, V. Egorychev31, S. Eidelman34, S. Eisenhardt50, U.
Eitschberger9, R. Ekelhof9, L. Eklund51,38, I. El Rifai5, Ch. Elsasser40, S.
Esen11, T. Evans55, A. Falabella16,f, C. Färber11, C. Farinelli41, S. Farry52,
D. Ferguson50, V. Fernandez Albor37, F. Ferreira Rodrigues1, M. Ferro-Luzzi38,
S. Filippov33, M. Fiore16,f, M. Fiorini16,f, M. Firlej27, C. Fitzpatrick38, T.
Fiutowski27, M. Fontana10, F. Fontanelli19,j, R. Forty38, O. Francisco2, M.
Frank38, C. Frei38, M. Frosini17,38,g, J. Fu21, E. Furfaro24,l, A. Gallas
Torreira37, D. Galli14,d, S. Gambetta19,j, M. Gandelman2, P. Gandini59, Y.
Gao3, J. Garofoli59, J. Garra Tico47, L. Garrido36, C. Gaspar38, R. Gauld55,
L. Gavardi9, E. Gersabeck11, M. Gersabeck54, T. Gershon48, Ph. Ghez4, A.
Gianelle22, S. Giani’39, V. Gibson47, L. Giubega29, V.V. Gligorov38, C.
Göbel60, D. Golubkov31, A. Golutvin53,31,38, A. Gomes1,a, H. Gordon38, C.
Gotti20, M. Grabalosa Gándara5, R. Graciani Diaz36, L.A. Granado Cardoso38, E.
Graugés36, G. Graziani17, A. Grecu29, E. Greening55, S. Gregson47, P.
Griffith45, L. Grillo11, O. Grünberg62, B. Gui59, E. Gushchin33, Yu. Guz35,38,
T. Gys38, C. Hadjivasiliou59, G. Haefeli39, C. Haen38, S.C. Haines47, S.
Hall53, B. Hamilton58, T. Hampson46, X. Han11, S. Hansmann-Menzemer11, N.
Harnew55, S.T. Harnew46, J. Harrison54, T. Hartmann62, J. He38, T. Head38, V.
Heijne41, K. Hennessy52, P. Henrard5, L. Henry8, J.A. Hernando Morata37, E.
van Herwijnen38, M. Heß62, A. Hicheur1, D. Hill55, M. Hoballah5, C. Hombach54,
W. Hulsbergen41, P. Hunt55, N. Hussain55, D. Hutchcroft52, D. Hynds51, M.
Idzik27, P. Ilten56, R. Jacobsson38, A. Jaeger11, J. Jalocha55, E. Jans41, P.
Jaton39, A. Jawahery58, M. Jezabek26, F. Jing3, M. John55, D. Johnson55, C.R.
Jones47, C. Joram38, B. Jost38, N. Jurik59, M. Kaballo9, S. Kandybei43, W.
Kanso6, M. Karacson38, T.M. Karbach38, M. Kelsey59, I.R. Kenyon45, T. Ketel42,
B. Khanji20, C. Khurewathanakul39, S. Klaver54, O. Kochebina7, M. Kolpin11, I.
Komarov39, R.F. Koopman42, P. Koppenburg41,38, M. Korolev32, A. Kozlinskiy41,
L. Kravchuk33, K. Kreplin11, M. Kreps48, G. Krocker11, P. Krokovny34, F.
Kruse9, M. Kucharczyk20,26,38,k, V. Kudryavtsev34, K. Kurek28, T.
Kvaratskheliya31, V.N. La Thi39, D. Lacarrere38, G. Lafferty54, A. Lai15, D.
Lambert50, R.W. Lambert42, E. Lanciotti38, G. Lanfranchi18, C. Langenbruch38,
B. Langhans38, T. Latham48, C. Lazzeroni45, R. Le Gac6, J. van Leerdam41,
J.-P. Lees4, R. Lefèvre5, A. Leflat32, J. Lefrançois7, S. Leo23, O. Leroy6, T.
Lesiak26, B. Leverington11, Y. Li3, M. Liles52, R. Lindner38, C. Linn38, F.
Lionetto40, B. Liu15, G. Liu38, S. Lohn38, I. Longstaff51, I. Longstaff51,
J.H. Lopes2, N. Lopez-March39, P. Lowdon40, H. Lu3, D. Lucchesi22,q, H. Luo50,
A. Lupato22, E. Luppi16,f, O. Lupton55, F. Machefert7, I.V. Machikhiliyan31,
F. Maciuc29, O. Maev30, S. Malde55, G. Manca15,e, G. Mancinelli6, M.
Manzali16,f, J. Maratas5, J.F. Marchand4, U. Marconi14, C. Marin Benito36, P.
Marino23,s, R. Märki39, J. Marks11, G. Martellotti25, A. Martens8, A. Martín
Sánchez7, M. Martinelli41, D. Martinez Santos42, F. Martinez Vidal64, D.
Martins Tostes2, A. Massafferri1, R. Matev38, Z. Mathe38, C. Matteuzzi20, A.
Mazurov16,38,f, M. McCann53, J. McCarthy45, A. McNab54, R. McNulty12, B.
McSkelly52, B. Meadows57,55, F. Meier9, M. Meissner11, M. Merk41, D.A.
Milanes8, M.-N. Minard4, J. Molina Rodriguez60, S. Monteil5, D. Moran54, M.
Morandin22, P. Morawski26, A. Mordà6, M.J. Morello23,s, J. Moron27, R.
Mountain59, F. Muheim50, K. Müller40, R. Muresan29, B. Muster39, P. Naik46, T.
Nakada39, R. Nandakumar49, I. Nasteva1, M. Needham50, N. Neri21, S. Neubert38,
N. Neufeld38, M. Neuner11, A.D. Nguyen39, T.D. Nguyen39, C. Nguyen-Mau39,p, M.
Nicol7, V. Niess5, R. Niet9, N. Nikitin32, T. Nikodem11, A. Novoselov35, A.
Oblakowska-Mucha27, V. Obraztsov35, S. Oggero41, S. Ogilvy51, O. Okhrimenko44,
R. Oldeman15,e, G. Onderwater65, M. Orlandea29, J.M. Otalora Goicochea2, P.
Owen53, A. Oyanguren64, B.K. Pal59, A. Palano13,c, F. Palombo21,t, M.
Palutan18, J. Panman38, A. Papanestis49,38, M. Pappagallo51, C. Parkes54, C.J.
Parkinson9, G. Passaleva17, G.D. Patel52, M. Patel53, C. Patrignani19,j, A.
Pazos Alvarez37, A. Pearce54, A. Pellegrino41, M. Pepe Altarelli38, S.
Perazzini14,d, E. Perez Trigo37, P. Perret5, M. Perrin-Terrin6, L.
Pescatore45, E. Pesen66, K. Petridis53, A. Petrolini19,j, E. Picatoste
Olloqui36, B. Pietrzyk4, T. Pilař48, D. Pinci25, A. Pistone19, S. Playfer50,
M. Plo Casasus37, F. Polci8, A. Poluektov48,34, E. Polycarpo2, A. Popov35, D.
Popov10, B. Popovici29, C. Potterat2, A. Powell55, J. Prisciandaro39, A.
Pritchard52, C. Prouve46, V. Pugatch44, A. Puig Navarro39, G. Punzi23,r, W.
Qian4, B. Rachwal26, J.H. Rademacker46, B. Rakotomiaramanana39, M. Rama18,
M.S. Rangel2, I. Raniuk43, N. Rauschmayr38, G. Raven42, S. Reichert54, M.M.
Reid48, A.C. dos Reis1, S. Ricciardi49, A. Richards53, K. Rinnert52, V. Rives
Molina36, D.A. Roa Romero5, P. Robbe7, A.B. Rodrigues1, E. Rodrigues54, P.
Rodriguez Perez54, S. Roiser38, V. Romanovsky35, A. Romero Vidal37, M.
Rotondo22, J. Rouvinet39, T. Ruf38, F. Ruffini23, H. Ruiz36, P. Ruiz Valls64,
G. Sabatino25,l, J.J. Saborido Silva37, N. Sagidova30, P. Sail51, B.
Saitta15,e, V. Salustino Guimaraes2, C. Sanchez Mayordomo64, B. Sanmartin
Sedes37, R. Santacesaria25, C. Santamarina Rios37, E. Santovetti24,l, M.
Sapunov6, A. Sarti18,m, C. Satriano25,n, A. Satta24, M. Savrie16,f, D.
Savrina31,32, M. Schiller42, H. Schindler38, M. Schlupp9, M. Schmelling10, B.
Schmidt38, O. Schneider39, A. Schopper38, M.-H. Schune7, R. Schwemmer38, B.
Sciascia18, A. Sciubba25, M. Seco37, A. Semennikov31, K. Senderowska27, I.
Sepp53, N. Serra40, J. Serrano6, L. Sestini22, P. Seyfert11, M. Shapkin35, I.
Shapoval16,43,f, Y. Shcheglov30, T. Shears52, L. Shekhtman34, V. Shevchenko63,
A. Shires9, R. Silva Coutinho48, G. Simi22, M. Sirendi47, N. Skidmore46, T.
Skwarnicki59, N.A. Smith52, E. Smith55,49, E. Smith53, J. Smith47, M. Smith54,
H. Snoek41, M.D. Sokoloff57, F.J.P. Soler51, F. Soomro39, D. Souza46, B. Souza
De Paula2, B. Spaan9, A. Sparkes50, F. Spinella23, P. Spradlin51, F. Stagni38,
S. Stahl11, O. Steinkamp40, O. Stenyakin35, S. Stevenson55, S. Stoica29, S.
Stone59, B. Storaci40, S. Stracka23,38, M. Straticiuc29, U. Straumann40, R.
Stroili22, V.K. Subbiah38, L. Sun57, W. Sutcliffe53, K. Swientek27, S.
Swientek9, V. Syropoulos42, M. Szczekowski28, P. Szczypka39,38, D. Szilard2,
T. Szumlak27, S. T’Jampens4, M. Teklishyn7, G. Tellarini16,f, E. Teodorescu29,
F. Teubert38, C. Thomas55, E. Thomas38, J. van Tilburg41, V. Tisserand4, M.
Tobin39, S. Tolk42, L. Tomassetti16,f, D. Tonelli38, S. Topp-Joergensen55, N.
Torr55, E. Tournefier4, S. Tourneur39, M.T. Tran39, M. Tresch40, A.
Tsaregorodtsev6, P. Tsopelas41, N. Tuning41, M. Ubeda Garcia38, A. Ukleja28,
A. Ustyuzhanin63, U. Uwer11, V. Vagnoni14, G. Valenti14, A. Vallier7, R.
Vazquez Gomez18, P. Vazquez Regueiro37, C. Vázquez Sierra37, S. Vecchi16, J.J.
Velthuis46, M. Veltri17,h, G. Veneziano39, M. Vesterinen11, B. Viaud7, D.
Vieira2, M. Vieites Diaz37, X. Vilasis-Cardona36,o, A. Vollhardt40, D.
Volyanskyy10, D. Voong46, A. Vorobyev30, V. Vorobyev34, C. Voß62, H. Voss10,
J.A. de Vries41, R. Waldi62, C. Wallace48, R. Wallace12, J. Walsh23, S.
Wandernoth11, J. Wang59, D.R. Ward47, N.K. Watson45, A.D. Webber54, D.
Websdale53, M. Whitehead48, J. Wicht38, D. Wiedner11, G. Wilkinson55, M.P.
Williams45, M. Williams56, F.F. Wilson49, J. Wimberley58, J. Wishahi9, W.
Wislicki28, M. Witek26, G. Wormser7, S.A. Wotton47, S. Wright47, S. Wu3, K.
Wyllie38, Y. Xie61, Z. Xing59, Z. Xu39, Z. Yang3, X. Yuan3, O. Yushchenko35,
M. Zangoli14, M. Zavertyaev10,b, F. Zhang3, L. Zhang59, W.C. Zhang12, Y.
Zhang3, A. Zhelezov11, A. Zhokhov31, L. Zhong3, A. Zvyagin38.

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 Milano, Milano, Italy
22Sezione INFN di Padova, Padova, Italy
23Sezione INFN di Pisa, Pisa, Italy
24Sezione INFN di Roma Tor Vergata, Roma, Italy
25Sezione INFN di Roma La Sapienza, Roma, Italy
26Henryk Niewodniczanski Institute of Nuclear Physics Polish Academy of
Sciences, Kraków, Poland
27AGH - University of Science and Technology, Faculty of Physics and Applied
Computer Science, Kraków, Poland
28National Center for Nuclear Research (NCBJ), Warsaw, Poland
29Horia Hulubei National Institute of Physics and Nuclear Engineering,
Bucharest-Magurele, Romania
30Petersburg Nuclear Physics Institute (PNPI), Gatchina, Russia
31Institute of Theoretical and Experimental Physics (ITEP), Moscow, Russia
32Institute of Nuclear Physics, Moscow State University (SINP MSU), Moscow,
Russia
33Institute for Nuclear Research of the Russian Academy of Sciences (INR RAN),
Moscow, Russia
34Budker Institute of Nuclear Physics (SB RAS) and Novosibirsk State
University, Novosibirsk, Russia
35Institute for High Energy Physics (IHEP), Protvino, Russia
36Universitat de Barcelona, Barcelona, Spain
37Universidad de Santiago de Compostela, Santiago de Compostela, Spain
38European Organization for Nuclear Research (CERN), Geneva, Switzerland
39Ecole Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland
40Physik-Institut, Universität Zürich, Zürich, Switzerland
41Nikhef National Institute for Subatomic Physics, Amsterdam, The Netherlands
42Nikhef National Institute for Subatomic Physics and VU University Amsterdam,
Amsterdam, The Netherlands
43NSC Kharkiv Institute of Physics and Technology (NSC KIPT), Kharkiv, Ukraine
44Institute for Nuclear Research of the National Academy of Sciences (KINR),
Kyiv, Ukraine
45University of Birmingham, Birmingham, United Kingdom
46H.H. Wills Physics Laboratory, University of Bristol, Bristol, United
Kingdom
47Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom
48Department of Physics, University of Warwick, Coventry, United Kingdom
49STFC Rutherford Appleton Laboratory, Didcot, United Kingdom
50School of Physics and Astronomy, University of Edinburgh, Edinburgh, United
Kingdom
51School of Physics and Astronomy, University of Glasgow, Glasgow, United
Kingdom
52Oliver Lodge Laboratory, University of Liverpool, Liverpool, United Kingdom
53Imperial College London, London, United Kingdom
54School of Physics and Astronomy, University of Manchester, Manchester,
United Kingdom
55Department of Physics, University of Oxford, Oxford, United Kingdom
56Massachusetts Institute of Technology, Cambridge, MA, United States
57University of Cincinnati, Cincinnati, OH, United States
58University of Maryland, College Park, MD, United States
59Syracuse University, Syracuse, NY, United States
60Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), Rio de
Janeiro, Brazil, associated to 2
61Institute of Particle Physics, Central China Normal University, Wuhan,
Hubei, China, associated to 3
62Institut für Physik, Universität Rostock, Rostock, Germany, associated to 11
63National Research Centre Kurchatov Institute, Moscow, Russia, associated to
31
64Instituto de Fisica Corpuscular (IFIC), Universitat de Valencia-CSIC,
Valencia, Spain, associated to 36
65KVI - University of Groningen, Groningen, The Netherlands, associated to 41
66Celal Bayar University, Manisa, Turkey, associated to 38

aUniversidade Federal do Triângulo Mineiro (UFTM), Uberaba-MG, Brazil
bP.N. Lebedev Physical Institute, Russian Academy of Science (LPI RAS),
Moscow, Russia
cUniversità di Bari, Bari, Italy
dUniversità di Bologna, Bologna, Italy
eUniversità di Cagliari, Cagliari, Italy
fUniversità di Ferrara, Ferrara, Italy
gUniversità di Firenze, Firenze, Italy
hUniversità di Urbino, Urbino, Italy
iUniversità di Modena e Reggio Emilia, Modena, Italy
jUniversità di Genova, Genova, Italy
kUniversità di Milano Bicocca, Milano, Italy
lUniversità di Roma Tor Vergata, Roma, Italy
mUniversità di Roma La Sapienza, Roma, Italy
nUniversità della Basilicata, Potenza, Italy
oLIFAELS, La Salle, Universitat Ramon Llull, Barcelona, Spain
pHanoi University of Science, Hanoi, Viet Nam
qUniversità di Padova, Padova, Italy
rUniversità di Pisa, Pisa, Italy
sScuola Normale Superiore, Pisa, Italy
tUniversità degli Studi di Milano, Milano, Italy
uPolitecnico di Milano, Milano, Italy

##  1 Introduction

The decay ${{B}^{0}}\\!\rightarrow\phi{{K}^{*0}}$ ‡‡‡In this paper ${K}^{*0}$
is defined as $K^{*}(892)^{0}$ unless otherwise stated. has a branching
fraction of $(9.8\pm 0.6)\times 10^{-6}$ [1]. In the Standard Model it
proceeds mainly via the gluonic penguin diagram shown in Fig. 1. Studies of
observables related to $C\\!P$ violation in this decay probe contributions
from physics beyond the Standard Model in the penguin loop [2, 3, 4]. The
decay was first observed by the CLEO collaboration [5]. Subsequently,
branching fraction measurements and angular analyses have been reported by the
BaBar and Belle collaborations [6, 7, 8, 9, 10, 11].

Figure 1: Leading Feynman diagram for the
${{B}^{0}}\\!\rightarrow\phi{{K}^{*0}}$ decay.

The decay involves a spin-0 $B$-meson decaying into two spin-1 vector mesons
($B\rightarrow VV$). Due to angular momentum conservation there are only three
independent configurations of the final-state spin vectors, a longitudinal
component where in the ${{B}^{0}}$ rest frame both resonances are polarized in
their direction of motion, and two transverse components with collinear and
orthogonal polarizations. Angular analyses have shown that the longitudinal
and transverse components in this decay have roughly equal amplitudes. Similar
results are seen in other ${B}\rightarrow VV$ penguin transitions [12, 13, 14,
15]. This is in contrast to tree-level decays such as
${{B}^{0}}\rightarrow\rho^{+}\rho^{-}$, where the $V-A$ nature of the weak
interaction causes the longitudinal component to dominate. The different
behaviour of tree and penguin decays has attracted much theoretical attention,
with several explanations proposed such as large contributions from penguin
annihilation effects [16] or final-state interactions [17]. More recent
calculations based on QCD factorization [18, 19] are consistent with the data,
although with significant uncertainties.

In this paper, measurements of the polarization amplitudes, phases, $C\\!P$
asymmetries and triple-product asymmetries are presented. In the Standard
Model the $C\\!P$ and triple-product asymmetries are expected to be small and
were found to be consistent with zero by previous experiments [6, 7, 8, 9,
10]. The studies reported here are performed using $pp$ collision data,
corresponding to an integrated luminosity of $1.0~{}\mbox{\,fb}^{-1}$,
collected at a centre-of-mass energy of
$\sqrt{s}=7\,\mathrm{\,Te\kern-1.00006ptV}$ with the LHCb detector.

##  2 Analysis strategy

In this analysis the ${{B}^{0}}\\!\rightarrow\phi{{K}^{*0}}$ decay is studied,
where the $\phi$ and ${K}^{*0}$ mesons decay to $K^{+}K^{-}$ and
$K^{+}\pi^{-}$, respectively (the study of the charge conjugate ${\kern
1.79993pt\overline{\kern-1.79993ptB}{}}^{0}$ mode is implicitly assumed in
this paper). Angular momentum conservation, for this pseudoscalar to vector-
vector transition, allows three possible helicity configurations of the
vector-meson pair, with amplitudes denoted $H_{+1}$, $H_{-1}$ and $H_{0}$.
These can be written as a longitudinal polarization, $A_{0}$, and two
transverse polarizations, $A_{\perp}$ and $A_{\parallel}$,

$A_{0}=H_{0}\;,\hskip
28.45274ptA_{\perp}=\frac{H_{+1}-H_{-1}}{\sqrt{2}}\;\hskip
28.45274pt\mbox{and}\hskip
28.45274ptA_{\parallel}=\frac{H_{+1}+H_{-1}}{\sqrt{2}}\;.$ (1)

In addition to the dominant vector-vector (P-wave) amplitudes, there are
contributions where either the $K^{+}K^{-}$ or $K^{+}\pi^{-}$ pairs are
produced in a spin-0 (S-wave) state. These amplitudes are denoted
$A_{\textrm{S}}^{KK}$ and $A_{\textrm{S}}^{K\pi}$, respectively. Only the
relative phases of the amplitudes are physical observables. A phase convention
is chosen such that $A_{0}$ is real. The remaining amplitudes have magnitudes
and relative phases defined as

$A_{\parallel}=|A_{\parallel}|e^{i\delta_{\parallel}}\;,\hskip
2.84544ptA_{\perp}=|A_{\perp}|e^{i\delta_{\perp}}\;,\hskip
2.84544ptA_{\textrm{S}}^{K\pi}=|A_{\textrm{S}}^{K\pi}|e^{i\delta_{\textrm{S}}^{K\pi}}\hskip
2.84544pt\mbox{and}\hskip
2.84544ptA_{\textrm{S}}^{KK}=|A_{\textrm{S}}^{KK}|e^{i\delta_{\textrm{S}}^{KK}}\;.$
(2)

To determine these quantities, an analysis of the angular distributions and
invariant masses of the decay products is performed. It is assumed that the
contribution from $B^{0}\rightarrow K^{+}K^{-}K^{+}\pi^{-}$, where both the
$K^{+}K^{-}$ and $K^{+}\pi^{-}$ are non-resonant, is negligible.

In the following sections the key elements of the analysis are discussed.
First, the conventions used in the angular analysis are defined together with
the form of the differential cross-section. Next, the parameterization of the
$K^{+}\pi^{-}$ and $K^{+}K^{-}$ mass distributions is discussed. Finally, the
triple-product asymmetries that can be derived from the angular variables are
defined.

###  2.1 Angular analysis

The angular analysis is performed in terms of three helicity angles
($\theta_{1},\theta_{2},\Phi$), as depicted in Fig. 2. The angle $\theta_{1}$
is defined as the angle between the ${K}^{+}$ direction and the reverse of the
${B}^{0}$ direction in the ${K}^{*0}$ rest frame. Similarly, $\theta_{2}$ is
the angle between the ${K}^{+}$ direction and the reverse of the ${B}^{0}$
direction in the $\phi$ rest frame. The angle $\Phi$ is the angle between the
decay planes of the $\phi$ and ${K}^{*0}$ mesons in the ${B}^{0}$ rest frame.

Figure 2:  The helicity angles $\theta_{1}$, $\theta_{2}$, $\Phi$ for the
${{B}^{0}}\\!\rightarrow\phi{{K}^{*0}}$ decay.

The flavour of the decaying ${B}^{0}$ meson is determined by the charge of the
kaon from the ${K}^{*0}$ decay. To determine the polarization amplitudes, the
${B}^{0}$ and ${\kern 1.79993pt\overline{\kern-1.79993ptB}{}}^{0}$ decays are
combined. For the study of $C\\!P$ asymmetries, the ${B}^{0}$ and ${\kern
1.79993pt\overline{\kern-1.79993ptB}{}}^{0}$ decays are separated.

Taking into account both the P- and S-wave contributions and their
interference, the differential decay rate [8] is given by the sum of the
fifteen terms given in Table 1,

$d^{5}\Gamma=\frac{9}{8\pi}\sum^{15}_{i=1}h_{i}~{}f_{i}(\theta_{1},\theta_{2},\Phi)\mathcal{M}_{i}(m_{K\pi},m_{KK})d\Omega(KKK\pi)\;.$
(3)

The $h_{i}$ factors are combinations of the amplitudes, $f_{i}$ are functions
of the helicity angles, $\mathcal{M}_{i}$ are functions of the invariant mass
of the intermediate resonances and $d\Omega(KKK\pi)$ is a four-body phase-
space factor,

$d\Omega(KKK\pi)\propto q_{\phi}q_{K^{*}}q_{{{B}^{0}}}\hskip
1.42262ptdm_{K\pi}\hskip 1.42262ptdm_{KK}\hskip 1.42262ptd\textrm{cos}\hskip
1.42262pt{\theta_{1}}\hskip 1.42262ptd\textrm{cos}\hskip
1.42262pt{\theta_{2}}\hskip 1.42262ptd\Phi\;,$ (4)

where $q_{A}$ is the momentum of the daughter particles in the mother’s ($A$ =
${{B}^{0}},\phi,{{K}^{*0}}$) centre-of-mass system.

Table 1: Definition of the $h_{i}$, $f_{i}$ and $\mathcal{M}_{i}$ terms in Eq.
3. Note that the P-wave interference terms $i=4$ and $i=6$ take the imaginary
parts of $A_{\perp}A_{\parallel}^{*}$ and $A_{\perp}A_{0}^{*}$, while $i=5$
takes the real part of $A_{\parallel}A_{0}^{*}$. Similarly the S-wave
interference terms $i=9$ and $i=13$ take the imaginary parts of
$A_{\perp}A_{\textrm{S}}^{*}M_{1}M_{0}^{*}$, and the terms $i=8,10,12,14$ take
the real parts of $A_{\parallel}A_{\textrm{S}}^{*}M_{1}M_{0}^{*}$ and
$A_{0}A_{\textrm{S}}^{*}M_{1}M_{0}^{*}$.

$i$ | $h_{i}$ | $f_{i}(\theta_{1},\theta_{2},\Phi)$ | $\mathcal{M}_{i}(m_{K\pi},m_{KK})$
---|---|---|---
1 | $|A_{0}|^{2}$ | $\cos\theta_{1}^{2}\cos\theta_{2}^{2}$ | $|M_{1}^{K\pi}(m_{K\pi})|^{2}|M_{1}^{KK}(m_{KK})|^{2}$
2 | $|A_{\parallel}|^{2}$ | $\frac{1}{4}\sin\theta_{1}^{2}\sin\theta_{2}^{2}(1+\cos(2\Phi))$ | $|M_{1}^{K\pi}(m_{K\pi})|^{2}|M_{1}^{KK}(m_{KK})|^{2}$
3 | $|A_{\perp}|^{2}$ | $\frac{1}{4}\sin\theta_{1}^{2}\sin\theta_{2}^{2}(1-\cos(2\Phi))$ | $|M_{1}^{K\pi}(m_{K\pi})|^{2}|M_{1}^{KK}(m_{KK})|^{2}$
4 | $|A_{\perp}||A_{\parallel}^{*}|e^{i(\delta_{\perp}-\delta_{\parallel})}$ | $-\frac{1}{2}\sin\theta_{1}^{2}\sin\theta_{2}^{2}\sin(2\Phi)$ | $|M_{1}^{K\pi}(m_{K\pi})|^{2}|M_{1}^{KK}(m_{KK})|^{2}$
5 | $|A_{\parallel}||A_{0}^{*}|e^{i\delta_{\parallel}}$ | $\sqrt{2}\cos\theta_{1}\sin\theta_{1}\cos\theta_{2}\sin\theta_{2}\cos\Phi$ | $|M_{1}^{K\pi}(m_{K\pi})|^{2}|M_{1}^{KK}(m_{KK})|^{2}$
6 | $|A_{\perp}||A_{0}^{*}|e^{i\delta_{\perp}}$ | $-\sqrt{2}\cos\theta_{1}\sin\theta_{1}\cos\theta_{2}\sin\theta_{2}\sin\Phi$ | $|M_{1}^{K\pi}(m_{K\pi})|^{2}|M_{1}^{KK}(m_{KK})|^{2}$
7 | $|A_{\textrm{S}}^{K\pi}|^{2}$ | $\frac{1}{3}\cos\theta_{2}^{2}$ | $|M_{0}^{K\pi}(m_{K\pi})|^{2}|M_{1}^{KK}(m_{KK})|^{2}$
8 | $|A_{\parallel}||A_{\textrm{S}}^{*K\pi}|e^{i(\delta_{\parallel}-\delta_{\textrm{S}}^{K\pi})}$ | $\frac{\sqrt{6}}{3}\sin\theta_{1}\cos\theta_{2}\sin\theta_{2}\cos\Phi$ | $|M_{1}^{KK}(m_{KK})|^{2}M_{1}^{K\pi}(m_{K\pi})M_{0}^{*K\pi}(m_{K\pi})$
9 | $|A_{\perp}||A_{\textrm{S}}^{*K\pi}|e^{i(\delta_{\perp}-\delta_{\textrm{S}}^{K\pi})}$ | $-\frac{\sqrt{6}}{3}\sin\theta_{1}\cos\theta_{2}\sin\theta_{2}\sin\Phi$ | $|M_{1}^{KK}(m_{KK})|^{2}M_{1}^{K\pi}(m_{K\pi})M_{0}^{*K\pi}(m_{K\pi})$
10 | $|A_{0}||A_{\textrm{S}}^{*K\pi}|e^{-i\delta_{\textrm{S}}^{K\pi}}$ | $\frac{2}{\sqrt{3}}\cos\theta_{1}\cos\theta_{2}^{2}$ | $|M_{1}^{KK}(m_{KK})|^{2}M_{1}^{K\pi}(m_{K\pi})M_{0}^{*K\pi}(m_{K\pi})$
11 | $|A_{\textrm{S}}^{KK}|^{2}$ | $\frac{1}{3}\cos\theta_{1}^{2}$ | $|M_{0}^{KK}(m_{KK})|^{2}|M_{1}^{K\pi}(m_{K\pi})|^{2}$
12 | $|A_{\parallel}||A_{\textrm{S}}^{*KK}|e^{i(\delta_{\parallel}-\delta_{\textrm{S}}^{KK})}$ | $\frac{\sqrt{6}}{3}\sin\theta_{1}\cos\theta_{1}\sin\theta_{2}\cos\Phi$ | $|M_{1}^{K\pi}(m_{K\pi})|^{2}M_{1}^{KK}(m_{KK})M_{0}^{*KK}(m_{KK})$
13 | $|A_{\perp}||A_{\textrm{S}}^{*KK}|e^{i(\delta_{\perp}-\delta_{\textrm{S}}^{KK})}$ | $-\frac{\sqrt{6}}{3}\sin\theta_{1}\cos\theta_{1}\sin\theta_{2}\sin\Phi$ | $|M_{1}^{K\pi}(m_{K\pi})|^{2}M_{1}^{KK}(m_{KK})M_{0}^{*KK}(m_{KK})$
14 | $|A_{0}||A_{\textrm{S}}^{*KK}|e^{-i\delta_{\textrm{S}}^{KK}}$ | $\frac{2}{\sqrt{3}}\cos\theta_{1}^{2}\cos\theta_{2}$ | $|M_{1}^{K\pi}(m_{K\pi})|^{2}M_{1}^{KK}(m_{KK})M_{0}^{*KK}(m_{KK})$
15 | $|A_{\textrm{S}}^{K\pi}||A_{\textrm{S}}^{*KK}|e^{i(\delta_{\textrm{S}}^{K\pi}-\delta_{\textrm{S}}^{KK})}$ | $\frac{2}{3}\cos\theta_{1}\cos\theta_{2}$ | $M_{1}^{KK}(m_{KK})M_{0}^{K\pi}(m_{K\pi})M_{0}^{*KK}(m_{KK})M_{1}^{*K\pi}(m_{K\pi})$

The differential decay rate for ${{\kern
1.79993pt\overline{\kern-1.79993ptB}{}}^{0}}\\!\rightarrow\phi{{\kern
1.99997pt\overline{\kern-1.99997ptK}{}}^{*0}}$ is obtained by defining the
angles using the charge conjugate final state particles and multiplying the
interference terms $f_{4},f_{6},f_{9},f_{13}$ by $-1$. To allow for direct
$C\\!P$ violation, the amplitudes $A_{j}$ are replaced by $\overline{A}_{j}$,
for $j={0,\parallel,\perp,\textrm{S}}$. The rate is normalized separately for
the ${\kern 1.79993pt\overline{\kern-1.79993ptB}{}}^{0}$ and ${B}^{0}$ decays
such that the P- and S-wave fractions are

$F_{\textrm{P}}=|A_{0}|^{2}+|A_{\parallel}|^{2}+|A_{\perp}|^{2}\;,\hskip
28.45274ptF_{\textrm{S}}=|A_{\textrm{S}}^{K\pi}|^{2}+|A_{\textrm{S}}^{KK}|^{2}\;,\hskip
28.45274ptF_{\textrm{P}}+F_{\textrm{S}}=1\;,$ (5)

and

$\overline{F}_{\textrm{P}}=|{\overline{A}_{0}}|^{2}+|{\overline{A}_{\parallel}}|^{2}+|{\overline{A}_{\perp}}|^{2}\;,\hskip
28.45274pt\overline{F}_{\textrm{S}}=|{\overline{A}_{\textrm{S}}^{K\pi}}|^{2}+|{\overline{A}_{\textrm{S}}^{KK}}|^{2}\;,\hskip
28.45274pt\overline{F}_{\textrm{P}}+\overline{F}_{\textrm{S}}=1\;.$ (6)

In addition, a convention is adopted such that the phases
$\delta_{\textrm{S}}^{K\pi}$ and $\delta_{\textrm{S}}^{KK}$ are defined as the
difference between the P- and S-wave phases at the ${K}^{*0}$ and $\phi$ meson
poles, respectively.

###  2.2 Mass distributions

The differential decay width depends on the invariant masses of the
$K^{+}\pi^{-}$ and $K^{+}K^{-}$ systems, denoted $m_{K\pi}$ and $m_{KK}$,
respectively. The P-wave $K^{+}\pi^{-}$ amplitude is parameterized using a
relativistic spin-$1$ Breit-Wigner resonance function,

$M^{K\pi}_{1}(m_{K\pi})=\frac{m_{K\pi}}{q_{K^{*}}}\hskip
2.84526pt\frac{m_{0}^{K^{*}}\Gamma_{1}^{K\pi}(m_{K\pi})}{(m_{0}^{K^{*}})^{2}-m_{K\pi}^{2}-im_{0}^{K^{*}}\Gamma_{1}^{K\pi}(m_{K\pi})}\;,$
(7)

where $m_{0}^{K^{*}}=895.81\,{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ [1] is
the ${K}^{*0}$ mass. The mass-dependent width is given by

$\Gamma_{1}^{K\pi}(m_{K\pi})=\Gamma_{0}^{K^{*}}\frac{m_{0}^{K^{*}}}{m_{K\pi}}\hskip
2.84526pt\frac{1+r^{2}q_{0}^{2}}{1+r^{2}q_{K^{*}}^{2}}\left(\frac{q_{K^{*}}}{q_{0}}\right)^{3}\;,$
(8)

where $q_{0}$ is the value of $q_{K^{*}}$ at $m_{0}^{K^{*}}$, $r=3.4~{}\hbar
c/$GeV [20] is the interaction radius and
$\Gamma_{0}^{K^{*}}=47.4{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ is the
natural width of the ${K}^{*0}$ meson [1]. The P-wave $K^{+}K^{-}$ amplitude,
denoted $M_{1}^{KK}(m_{KK})$, is modelled in a similar way using the values
$m_{0}^{\phi}=1019.455\,{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ and
$\Gamma_{0}^{\phi}=4.26\,{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ [1]. In the
case of the $\phi$ meson the natural width is comparable to the detector
resolution of $1.2\,{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$, which is
accounted for by convolving the Breit-Wigner with a Gaussian function.

As the ${K}^{*0}$ is a relatively broad resonance, the S-wave component in the
$K^{+}\pi^{-}$ system, denoted $M_{0}^{K\pi}(m_{K\pi})$, needs careful
treatment. In this analysis the approach described in Ref. [8] is followed,
which makes use of the LASS parameterization [20]. This takes into account an
$L=0$ $K^{*}_{0}(1430)$ contribution together with a non-resonant amplitude.
The values used for the LASS parameterization are taken from Ref. [8].

Finally, an S-wave in the $K^{+}K^{-}$ system is considered. This is described
by the Flatté parameterization of the $f_{0}(980)$ resonance[21],

$M_{0}^{KK}(m_{KK})=\frac{1}{m_{f_{0}}^{2}-m_{KK}^{2}-im_{f_{0}}(g_{\pi\pi}\rho_{\pi\pi}+g_{KK}\rho_{KK})}\;,$
(9)

where the $g_{KK,\pi\pi}$ are partial decay widths and the $\rho_{KK,\pi\pi}$
are phase-space factors. The values
$m_{f_{0}}=939{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$,
$g_{\pi\pi}=199{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ and
$g_{KK}/g_{\pi\pi}=3.0$ were measured in Ref. [22]. The Flatté distribution is
convolved with a Gaussian function to account for the detector resolution.
Other approaches to modelling the mass distributions for both the
$K^{+}\pi^{-}$ and $K^{+}K^{-}$ S-wave are considered as part of the
systematic uncertainty determination.

###  2.3 Triple-product asymmetries

The amplitudes and phases can be used to calculate triple-product asymmetries
[4, 23, 2]. Non-zero triple-product asymmetries arise either due to a
$T$-violating phase or a $C\\!P$-conserving phase and final-state
interactions. Assuming $C\\!PT$ symmetry, a $T$-violating phase, which is a
true asymmetry, implies that $C\\!P$ is violated.

For the P-wave decay, two triple-product asymmetries are calculated from the
results of the angular analysis [4],

$A_{T}^{1}=\frac{\Gamma(s_{\theta_{1}\theta_{2}}\sin\Phi>0)-\Gamma(s_{\theta_{1}\theta_{2}}\sin\Phi<0)}{\Gamma(s_{\theta_{1}\theta_{2}}\sin\Phi>0)+\Gamma(s_{\theta_{1}\theta_{2}}\sin\Phi<0)}\hskip
11.38092pt\mbox{and}\hskip 11.38092ptA_{T}^{2}=\frac{\Gamma(\sin
2\Phi>0)-\Gamma(\sin 2\Phi<0)}{\Gamma(\sin 2\Phi>0)+\Gamma(\sin 2\Phi<0)}\;,$
(10)

where $s_{\theta_{1}\theta_{2}}=\textrm{sign}(\cos\theta_{1}\cos\theta_{2})$.
These asymmetries can be rewritten in terms of the interference terms between
the amplitudes [4], $h_{4}$ and $h_{6}$ in Table 1,

$A_{T}^{1}=-\frac{4}{\pi}\mathcal{I}m(A_{\perp}A_{0}^{*})\hskip
28.45274pt\mbox{and}\hskip
28.45274ptA_{T}^{2}=-\frac{2\sqrt{2}}{\pi}\mathcal{I}m(A_{\perp}A_{\parallel}^{*})\;.$
(11)

Since the decay products identify the flavour at decay, the data can be
separated into ${{B}^{0}}$ and ${{\kern
1.79993pt\overline{\kern-1.79993ptB}{}}^{0}}$ decays and the triple-product
asymmetries calculated for both cases. This allows a determination of the true
asymmetries, $A_{T}^{\,k}{\rm(true)}=({A^{k}_{T}+\overline{A}^{\,k}_{T}})/2$,
and so called fake asymmetries,
$A_{T}^{k}{\rm(fake)}=(A^{k}_{T}-\overline{A}^{\,k}_{T})/2$, where $k=1,2$. In
the Standard Model the value of $A_{T}^{k}$(true) is predicted to be zero and
any deviation from this would indicate physics beyond the Standard Model. Non-
zero values for $A_{T}^{k}$(fake) reflect the importance of strong final-state
phases [4].

The S-wave contributions allow two additional triple-product asymmetries to be
defined from $h_{9}$ and $h_{13}$ in Table 1,

$\displaystyle A_{T}^{3}$
$\displaystyle=\frac{\displaystyle\Gamma(s_{\theta_{1}}\sin\Phi>0)-\Gamma(s_{\theta_{1}}\sin\Phi<0)}{\Gamma(s_{\theta_{1}}\sin\Phi>0)+\Gamma(s_{\theta_{1}}\sin\Phi<0)}$
$\displaystyle=-\sqrt{\frac{3}{2}}\int|M_{1}^{KK}(m_{KK})|^{2}\mathcal{I}m(A_{\perp}A_{\textrm{S}}^{*K\pi}M_{1}^{K\pi}(m_{K\pi})M_{0}^{*K\pi}(m_{K\pi}))dm_{KK}dm_{K\pi}\;,$
(12)

and

$\displaystyle A_{T}^{4}$
$\displaystyle=\frac{\Gamma(s_{\theta_{2}}\sin\Phi>0)-\Gamma(s_{\theta_{2}}\sin\Phi<0)}{\Gamma(s_{\theta_{2}}\sin\Phi>0)+\Gamma(s_{\theta_{2}}\sin\Phi<0)}$
$\displaystyle=-\sqrt{\frac{3}{2}}\int|M_{1}^{K\pi}(m_{K\pi})|^{2}\mathcal{I}m(A_{\perp}A_{\textrm{S}}^{*KK}M_{1}^{KK}(m_{KK})M_{0}^{*KK}(m_{KK}))dm_{KK}dm_{K\pi}\;,$
(13)

where $s_{\theta_{i}}=\textrm{sign}(\cos\theta_{i})$ for $i=1,2$.

##  3 Detector and dataset

The LHCb detector [24] is a single-arm forward spectrometer covering the
pseudorapidity range $2<\eta<5$, designed for the study of particles
containing $b$ or $c$ quarks. The detector includes a high-precision tracking
system consisting of a silicon-strip vertex detector surrounding the $pp$
interaction region, a large-area silicon-strip detector located upstream of a
dipole magnet with a bending power of about $4{\rm\,Tm}$, and three stations
of silicon-strip detectors and straw drift tubes placed downstream. The
polarity of the dipole magnet is reversed at intervals corresponding to
roughly $0.1\mbox{\,fb}^{-1}$ of collected data, in order to minimize
systematic uncertainties associated with detector asymmetries. The combined
tracking system provides a momentum measurement with relative uncertainty that
varies from 0.4 % at 5${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ to 0.6 % at
100${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$, and impact parameter resolution of
20${\,\upmu\rm m}$ for tracks with high transverse momentum ($p_{\rm T}$).
Charged hadrons are identified using two ring-imaging Cherenkov detectors
[25]. Photon, electron and hadron candidates are identified by a calorimeter
system consisting of scintillating-pad and preshower detectors, an
electromagnetic calorimeter and a hadronic calorimeter. Muons are identified
by a system composed of alternating layers of iron and multiwire proportional
chambers.

The trigger [26] consists of a hardware stage, based on information from the
calorimeter and muon systems, followed by a software stage, which applies a
full event reconstruction. In this analysis two categories of events that pass
the hardware trigger stage are considered: those where the signal $b$-hadron
products are used in the trigger decision (TOS) and those where the trigger
decision is caused by other activity in the event (TIS) [26]. The software
trigger requires a three-track secondary vertex with large transverse momenta
of the tracks and a significant displacement from the primary $pp$ interaction
vertices (PVs). At least one track should have $\mbox{$p_{\rm
T}$}>1.7{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ and $\chi^{2}_{\rm IP}$ with
respect to any primary interaction greater than 16, where $\chi^{2}_{\rm IP}$
is defined as the difference in $\chi^{2}$ of a given PV reconstructed with
and without the considered track. A multivariate algorithm [27] is used for
the identification of secondary vertices consistent with the decay of a $b$
hadron.

Simulated data samples are used to correct for the detector acceptance and
response. In the simulation, $pp$ collisions are generated using Pythia 6.4
[28] with a specific LHCb configuration [29]. Decays of hadronic particles are
described by EvtGen [30], in which final-state radiation is generated using
Photos [31]. The interaction of the generated particles with the detector and
its response are implemented using the Geant4 toolkit [32,
*Agostinelli:2002hh] as described in Ref. [34].

##  4 Event selection

The selection of events is divided into two parts. In the first step a loose
selection is performed that retains the majority of signal events, whilst
reducing the background by a large fraction. Following this, a multivariate
method is used to further reduce the background.

The selection starts from well reconstructed charged particles with a
$\mbox{$p_{\rm T}$}>500\,{\mathrm{\,Me\kern-1.00006ptV\\!/}c}$ that traverse
the entire spectrometer. Fake tracks, not associated to actual charged
particles, are suppressed using the output of a neural network trained to
discriminate between these and real particles [35]. Further background
suppression is achieved by exploiting the fact that the products of $b$-hadron
decays have a large impact parameter (IP) with respect to the nearest PV. The
IP of each track with respect to any primary vertex is required to have a
$\chi^{2}_{\rm IP}>9$.

To select well-identified pions and kaons, the difference in the logarithms of
the likelihood of the kaon hypothesis relative to the pion hypothesis
($\mathrm{DLL}_{{K}{\pi}}$) is provided using information from the ring-
imaging Cherenkov detectors. The kaons that form the $\phi\rightarrow
K^{+}K^{-}$ candidate are required to have $\mathrm{DLL}_{{K}{\pi}}>0$. To
reduce background from $\pi^{+}\pi^{-}$ pairs, a tighter requirement,
$\mathrm{DLL}_{{K}{\pi}}>2$, is applied to the kaon in the $K^{+}\pi^{-}$
pair. For the pion in the $K^{+}\pi^{-}$ pair the requirement is
$\mathrm{DLL}_{{K}{\pi}}<0$.

The resulting charged particles are combined to form $\phi$ and ${K}^{*0}$
meson candidates. The invariant mass of the $K^{+}K^{-}$ ($K^{+}\pi^{-})$ pair
is required to be within $\pm 15{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$
($\pm 150{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$) of the known mass of the
$\phi$ (${K}^{*0}$) meson [1]. Finally, the $p_{\rm T}$ of the $\phi$ and
${K}^{*0}$ mesons should both be greater than
$900{\mathrm{\,Me\kern-1.00006ptV\\!/}c}$, and the fit of their two-track
vertices should have a $\chi^{2}<9$.

Candidate ${B}^{0}$ meson decays with $K^{+}K^{-}K^{+}\pi^{-}$ invariant mass
in the range $5150<m_{KKK\pi}<5600{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$
are formed from pairs of selected $\phi$ and ${K}^{*0}$ meson candidates. A
fit is made requiring all four final-state particles to originate from a
common vertex and the $\chi^{2}$ per degree of freedom of this fit is required
to be less than 15. To remove ${{B}^{0}_{s}}\rightarrow\phi\phi$ decays where
a kaon has been incorrectly identified as a pion, the invariant mass of the
$K^{+}\pi^{-}$ pair is recalculated assuming that both particles are kaons. If
the resulting invariant mass is within $\pm
15{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ of the known $\phi$ mass, the
candidate is rejected. Finally, the decay vertex of the ${B}^{0}$ meson
candidate is required to be displaced from the nearest PV, with a flight
distance significance of more than 5 standard deviations, and the ${B}^{0}$
momentum vector is required to point back towards the PV with an impact
parameter less than $0.3$ mm and $\chi^{2}_{\rm IP}<5$.

Further background suppression is achieved using a geometric likelihood (GL)
method [36, 15, 37]. The GL is trained using a sample of simulated
${{B}^{0}}\\!\rightarrow\phi{{K}^{*0}}$ signal events together with background
events selected from the upper mass sideband of the ${B}^{0}$ meson,
$m_{KKK\pi}>5413{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$, and the $\phi$ mass
sidebands, $|m_{KK}-m_{0}^{\phi}|>15{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$.
These sidebands are not used in the subsequent analysis. Six discriminating
variables are input to the GL: the IP of the ${B}^{0}$ candidate with respect
to the PV, the distance of closest approach of the $\phi$ and $K^{*0}$ meson
candidate trajectories, the lifetime of the ${B}^{0}$ candidate, the
transverse momentum of the ${B}^{0}$ candidate, the minimum $\chi^{2}_{\rm
IP}$ of the $K^{+}K^{-}$ pair and the minimum $\chi^{2}_{\rm IP}$ of the
$K^{+}\pi^{-}$ pair. As a figure of merit the ratio $S/\sqrt{S+B}$ is
considered, where $S$ and $B$ are the yields of signal and background events
in the training samples, scaled to match the observed signal and background
yields in the data. The maximum value for the figure of merit is found to be
$24.6$ for $\textrm{GL}>0.1$, with signal and background efficiencies of
$90\,\%$ and $21\,\%$, respectively, compared to the selection performed
without the GL. This reduces the sample size for the final analysis to 1852
candidates.

##  5 ${K^{+}K^{-}K^{+}\pi^{-}}$ mass model

The signal yield is determined by an unbinned maximum likelihood fit to the
$K^{+}K^{-}K^{+}\pi^{-}$ invariant mass distribution. The selected mass range
is chosen to avoid modelling partially reconstructed $B$ decays with a missing
hadron or photon. In the fit the signal invariant mass distribution is
modelled as the sum of a Crystal Ball function [38] and a wider Gaussian
function with a common mean. The width and fraction of the Gaussian function
are fixed to values obtained using simulated events. A component is also
included to account for the small contribution from the decay ${{\kern
1.79993pt\overline{\kern-1.79993ptB}{}}^{0}_{s}}\\!\rightarrow\phi{{K}^{*0}}$
[37]. The shape parameters for this component are in common with the
${{B}^{0}}$ signal shape and the relative position of the ${{B}^{0}_{s}}$
signal with respect to the ${{B}^{0}}$ signal is fixed using the known mass
difference between ${B}^{0}$ and ${B}^{0}_{s}$ mesons [1]. The invariant mass
distribution is shown in Fig. 3, together with the result of the fit, from
which a yield of $1655\pm 42$ $B^{0}$ signal candidates is found.

After the selection the background is mainly combinatorial and is modelled by
an exponential. Background from ${{B}^{0}_{s}}\rightarrow\phi\phi$ decays,
with one of the kaons misidentified as a pion is reduced by the veto applied
in the selection. The number of candidates from this source is estimated to be
6 events using simulation. These are distributed across the
$K^{+}K^{-}K^{+}\pi^{-}$ mass range, and are considered negligible in the fit.
A potential background from ${{B}^{0}}\rightarrow
D_{s}^{+}K^{-}(D_{s}^{+}\rightarrow\phi\pi^{+})$ decays, which would peak in
the signal region, is also found to be negligible. Possible background from
the yet unobserved decay $\Lambda_{b}^{0}\rightarrow\phi{p}K^{-}$ with a
misidentified proton is considered as part of the systematic uncertainties.

Figure 3: Invariant mass distribution for selected $K^{+}K^{-}K^{+}\pi^{-}$
candidates. A fit to the model described in the text is superimposed (red
solid line). The signal contribution is shown as the blue dotted line. The
contribution from combinatorial background is shown in green (dotted line). A
contribution from ${{\kern
1.61993pt\overline{\kern-1.61993ptB}{}}^{0}_{s}}\\!\rightarrow\phi{{K}^{*0}}$
(purple dot-dashed line) decays is visible around the known $B_{s}^{0}$ meson
mass.

##  6 Angular fit

The physics parameters of interest for this analysis are defined in Table 2.
They include the polarization amplitudes, phases and amplitude differences
between ${B}^{0}$ and ${\kern 1.79993pt\overline{\kern-1.79993ptB}{}}^{0}$
decays from which the triple-product asymmetries are calculated.

The correlation between the fit variables and $m_{KKK\pi}$ is found to be less
than $3\,\%$. Therefore, the background can be subtracted using the sPlot
method [39], with $m_{K^{+}K^{-}K^{+}\pi^{-}}$ as the discriminating variable.
The results of the invariant mass fit discussed in Sec. 5 are used to give
each candidate a signal weight, $W_{n}$, which is a function of
$m_{K^{+}K^{-}K^{+}\pi^{-}}$. The weight is used to subtract the background
contributions from the distributions of the decay angles and intermediate
resonance masses, which are fit using a signal-only likelihood that is a
function of $\theta_{1},\theta_{2},\Phi,m_{K\pi}$ and $m_{KK}$. The angular
fit minimizes the negative log likelihood summed over the $n$ selected
candidates

$-\textrm{ln}\mathcal{L}=-\alpha\sum_{n}W_{n}\textrm{ln}\mathcal{S}_{n}\;,$
(14)

where $\alpha=\sum_{n}W_{n}/\sum_{n}W_{n}^{2}$ is a normalization factor that
includes the effect of the weights in the determination of the uncertainties
[40, 41], and $\mathcal{S}$ is the signal probability density function (Eq. 3)
convolved with the detector acceptance.

The acceptance of the detector is not uniform as a function of the decay angle
of the $K^{+}\pi^{-}$ system ($\theta_{1}$) and the $K^{+}\pi^{-}$ invariant
mass. This is due to the $500\,{\mathrm{\,Me\kern-1.00006ptV\\!/}c}$ criterion
applied on the $p_{\rm T}$ of the pion from the ${K}^{*0}$ meson decay. In
contrast, the acceptance is relatively uniform as a function of the decay
angles $\theta_{2}$ and $\Phi$, and the invariant mass of the $K^{+}K^{-}$
system.

The detector acceptance is modelled using a four-dimensional function that
depends on the three decay angles and the $K^{+}\pi^{-}$ invariant mass. The
shape of this function is obtained from simulated data. As the quantities
relating to the $p_{\rm T}$ of the decay products are used in the first-level
hardware based trigger, the acceptance is different for candidates that have a
TIS or TOS decision at the hardware trigger stage [26]. Consequently, the
trigger acceptance is calculated and corrected separately for the two
categories. The $17\,\%$ of candidates that fall in the overlap between the
two categories are treated as TOS, and the remaining TIS candidates are
labelled ‘not TOS’. The projections of the acceptance are shown in Fig. 4. In
the subsequent analysis the data set is divided into the two categories and a
simultaneous fit is performed.

Figure 4: Binned projections of the detector acceptance for (a) $m_{K\pi}$,
(b) $m_{KK}$ , (c) $\cos\theta_{1}$, (d) $\cos\theta_{2}$ and (e) $\Phi$. The
acceptance for the TOS (filled crosses) and not TOS (open squares) are shown
on each plot.

##  7 Angular analysis results

Figure 5 shows the data distribution for the intermediate resonance masses and
helicity angles with the projections of the best fit overlaid. The goodness of
fit is estimated using a point-to-point dissimilarity test [42], the
corresponding $p$-value is 0.64.

Figure 5: Data distribution for the helicity angles and of the intermediate
resonance masses: (a) $m_{K\pi}$ and (b) $m_{KK}$, (c) $\cos\theta_{1}$, (d)
$\cos\theta_{2}$ and (e) $\Phi$. The background has been subtracted using the
sPlot technique. The results of the fit are superimposed.

The fit results are listed in Table 2. The value of $f_{L}$ returned by the
fit is close to 0.5, indicating that the longitudinal and transverse
polarizations have similar size. Significant S-wave contributions are found in
both the $K^{+}\pi^{-}$ and $K^{+}K^{-}$ systems. The $C\\!P$ asymmetries in
both the amplitudes and the phases are consistent with zero.

Using Eqs. 11 – 13, the values for the triple-product asymmetries are derived
from the measured parameters and given in Table 3. The true asymmetries are
consistent with zero, showing no evidence for physics beyond the Standard
Model. In contrast, all but one of the fake asymmetries are significantly
different from zero, indicating the presence of final-state interactions.

The systematic uncertainties on the measured amplitudes, phases and triple-
product asymmetries are summarized in Table 4. The largest systematic
uncertainties on the results of the angular analysis arise from the
understanding of the detector acceptance. The angular acceptance function is
determined from simulated events as described in Sec. 6. An uncertainty,
labelled ‘Acceptance’ in the table, is assigned to account for the limited
size of the simulation sample used. This is estimated using pseudo-experiments
with a simplified simulation.

A difference is observed in the kinematic distributions of the final-state
particles between data and simulation. This is attributed to the S-wave
components, which are not included in the simulation. To account for this, the
simulated events are reweighted to match the signal distributions as expected
from the best estimate of the physics parameters from data (including the
S-wave). In addition, the events are reweighted to match the observed
distributions of the ${B}^{0}$ candidate and final-state particle transverse
momenta. The reweighting is done separately for the two trigger categories and
the nominal results are recalculated using the reweighted simulation to
determine the angular acceptance. The difference between the weighted and
unweighted results is taken as a systematic uncertainty (labelled ‘Data/MC’ in
the table).

A further uncertainty arises from the $K^{+}K^{-}K^{+}\pi^{-}$ mass model used
to determine the signal weights for the angular analysis. The fit procedure is
repeated using different signal and background models. For the signal
component a double Gaussian model is used instead of the sum of a Gaussian and
a Crystal Ball function. Similarly, the influence of background modelling is
probed using a first-order polynomial instead of an exponential function.
Other changes to the background model are related to the possible presence of
additional backgrounds. A possible small contribution from misidentified
$\Lambda_{b}\rightarrow pK^{-}K^{+}K^{-}$ and $\Lambda_{b}\rightarrow
p\pi^{-}K^{+}K^{+}$ decays is added and the fit repeated. Finally, the lower
bound of the fit range is varied and the contribution from partially
reconstructed $B$ decays modelled. The largest difference compared to the
central values is assigned as an estimate of the systematic uncertainty
(labelled ‘Mass model’ in the table).

Alternative models of the S-wave contributions in both the $K^{+}K^{-}$ and
$K^{+}\pi^{-}$ system are considered. The default fit uses the LASS
parameterization to model the $K^{+}\pi^{-}$ S-wave. As variations of this,
both a pure phase-space model and a spin-0 relativistic Breit-Wigner with mean
and width of the $K^{*}_{0}(1430)$ resonance are considered [1]. For the
$K^{+}K^{-}$ S-wave a pure phase-space model is tried in place of the Flatté
parameterization. The largest observed deviation from the nominal fit is taken
as a systematic uncertainty (column labelled ‘S-wave’ in the table).

Various consistency checks of the results are made. As a cross-check
candidates that are in the overlap between the trigger categories are treated
as TIS for the angular correction in the fit rather than TOS. The dataset is
also divided according to the magnetic field polarity. The results obtained in
these studies are consistent with the nominal results and no additional
uncertainty is assigned.

Table 2: Parameters measured in the angular analysis. The first and second uncertainties are statistical and systematic, respectively.  Parameter | Definition | Fitted value
---|---|---
$f_{\textrm{L}}$ | $0.5(|A_{0}|^{2}/F_{\textrm{P}}+|\overline{A}_{0}|^{2}/\overline{F}_{\textrm{P}})$ |  - $0.497\pm 0.019\pm 0.015$
$f_{\perp}$ | $0.5(|A_{\perp}|^{2}/F_{\textrm{P}}+|\overline{A}_{\perp}|^{2}/\overline{F}_{\textrm{P}})$ |  - $0.221\pm 0.016\pm 0.013$
$f_{\textrm{S}}(K\pi)$ | $0.5(|A_{\textrm{S}}^{K\pi}|^{2}+|\overline{A}_{\textrm{S}}^{K\pi}|^{2})$ |  - $0.143\pm 0.013\pm 0.012$
$f_{\textrm{S}}(KK)$ | $0.5(|A_{\textrm{S}}^{KK}|^{2}+|\overline{A}_{\textrm{S}}^{KK}|^{2})$ |  - $0.122\pm 0.013\pm 0.008$
$\delta_{\perp}$ | $0.5(\arg{A_{\perp}}+\arg{\overline{A}_{\perp}})$ |  - $2.633\pm 0.062\pm 0.037$
$\delta_{\parallel}$ | $0.5(\arg{A_{\parallel}}+\arg{\overline{A}_{\parallel}})$ |  - $2.562\pm 0.069\pm 0.040$
$\delta_{\textrm{S}}(K\pi)$ | $0.5(\arg{A_{\textrm{S}}^{K\pi}}+\arg{\overline{A}_{\textrm{S}}^{K\pi}})$ |  - $2.222\pm 0.063\pm 0.081$
$\delta_{\textrm{S}}(KK)$ | $0.5(\arg{A_{\textrm{S}}^{KK}}+\arg{\overline{A}_{\textrm{S}}^{KK}})$ |  - $2.481\pm 0.072\pm 0.048$
$\mathcal{A}_{0}^{{C\\!P}}$ | $(|A_{0}|^{2}{/F_{\textrm{P}}}-|\overline{A}_{0}|^{2}{/\overline{F}_{\textrm{P}}})/(|A_{0}|^{2}{/F_{\textrm{P}}}+|\overline{A}_{0}|^{2}{/\overline{F}_{\textrm{P}}})$ | $-0.003\pm 0.038\pm 0.005$
$\mathcal{A}_{\perp}^{{C\\!P}}$ | $(|A_{\perp}|^{2}{/F_{\textrm{P}}}-|\overline{A}_{\perp}|^{2}{/\overline{F}_{\textrm{P}}})/(|A_{\perp}|^{2}{/F_{\textrm{P}}}+|\overline{A}_{\perp}|^{2}{/\overline{F}_{\textrm{P}}})$ | $+0.047\pm 0.074\pm 0.009$
$\mathcal{A}_{S}(K\pi)^{{C\\!P}}$ | $(|A_{\textrm{S}}^{K\pi}|^{2}-|\overline{A}_{\textrm{S}}^{K\pi}|^{2})/(|A_{\textrm{S}}^{K\pi}|^{2}+|\overline{A}_{\textrm{S}}^{K\pi}|^{2})$ | $+0.073\pm 0.091\pm 0.035$
$\mathcal{A}_{S}(KK)^{{C\\!P}}$ | $(|A_{\textrm{S}}^{KK}|^{2}-|\overline{A}_{\textrm{S}}^{KK}|^{2})/(|A_{\textrm{S}}^{KK}|^{2}+|\overline{A}_{\textrm{S}}^{KK}|^{2})$ | $-0.209\pm 0.105\pm 0.012$
$\delta_{\perp}^{{C\\!P}}$ | $0.5(\arg{A_{\perp}}-\arg{\overline{A}_{\perp}})$ | $+0.062\pm 0.062\pm 0.005$
$\delta_{\parallel}^{{C\\!P}}$ | $0.5(\arg{A_{\parallel}}-\arg{\overline{A}_{\parallel}})$ | $+0.045\pm 0.069\pm 0.015$
$\delta_{S}(K\pi)^{{C\\!P}}$ | $0.5(\arg{A_{\textrm{S}}^{K\pi}}-\arg{\overline{A}_{\textrm{S}}^{K\pi}})$ | $+0.062\pm 0.062\pm 0.022$
$\delta_{S}(KK)^{{C\\!P}}$ | $0.5(\arg{A_{\textrm{S}}^{KK}}-\arg{\overline{A}_{\textrm{S}}^{KK}})$ | $+0.022\pm 0.072\pm 0.004$
Table 3: Triple-product asymmetries. The first and second uncertainties on the measured values are statistical and systematic, respectively.  Asymmetry | Measured value
---|---
$A_{T}^{1}$(true) | $-0.007\pm 0.012\pm 0.002$
$A_{T}^{2}$(true) | $+0.004\pm 0.014\pm 0.002$
$A_{T}^{3}$(true) | $+0.004\pm 0.006\pm 0.001$
$A_{T}^{4}$(true) | $+0.002\pm 0.006\pm 0.001$
$A_{T}^{1}$(fake) | $-0.105\pm 0.012\pm 0.006$
$A_{T}^{2}$(fake) | $-0.017\pm 0.014\pm 0.003$
$A_{T}^{3}$(fake) | $-0.063\pm 0.006\pm 0.005$
$A_{T}^{4}$(fake) | $-0.019\pm 0.006\pm 0.007$
Table 4: Systematic uncertainties on the measurement of the polarization amplitudes, relative strong phases and triple-product asymmetries. The column labelled ‘Total’ is the quadratic sum of the individual contributions.  Measurement | Acceptance | Data/MC | Mass model | S-wave | Total
---|---|---|---|---|---
$f_{\textrm{L}}$ | 0.014 | 0.005 | 0.002 | 0.001 | 0.015
$f_{\perp}$ | 0.013 | 0.002 | 0.001 | 0.001 | 0.013
$f_{\textrm{S}}(K\pi)$ | 0.012 | - | 0.001 | 0.002 | 0.012
$f_{\textrm{S}}(KK)$ | 0.007 | - | 0.002 | 0.003 | 0.008
$\delta_{\perp}$ | 0.023 | 0.010 | 0.006 | 0.026 | 0.037
$\delta_{\parallel}$ | 0.029 | 0.013 | 0.004 | 0.024 | 0.040
$\delta_{\textrm{S}}(K\pi)$ | 0.045 | 0.026 | 0.004 | 0.062 | 0.081
$\delta_{\textrm{S}}(KK)$ | 0.045 | 0.005 | 0.004 | 0.016 | 0.048
$A_{0}^{{C\\!P}}$ | - | 0.002 | 0.002 | 0.004 | 0.005
$A_{\perp}^{{C\\!P}}$ | - | 0.001 | 0.006 | 0.007 | 0.009
$A_{\textrm{S}}(K\pi)^{{C\\!P}}$ | - | 0.007 | 0.005 | 0.034 | 0.035
$A_{\textrm{S}}(KK)^{{C\\!P}}$ | - | 0.007 | 0.009 | 0.003 | 0.012
$\delta_{\perp}^{{C\\!P}}$ | - | 0.003 | 0.001 | 0.004 | 0.005
$\delta_{\parallel}^{{C\\!P}}$ | - | 0.005 | 0.002 | 0.014 | 0.015
$\delta_{\textrm{S}}(K\pi)^{{C\\!P}}$ | - | 0.005 | 0.003 | 0.021 | 0.022
$\delta_{\textrm{S}}(KK)^{{C\\!P}}$ | - | 0.002 | 0.002 | 0.003 | 0.004
$A_{T}^{1}$(true) | - | 0.0005 | 0.0005 | 0.002 | 0.002
$A_{T}^{2}$(true) | - | 0.0006 | 0.0005 | 0.002 | 0.002
$A_{T}^{3}$(true) | - | 0.0002 | 0.0003 | 0.001 | 0.001
$A_{T}^{4}$(true) | - | 0.0002 | 0.0003 | 0.001 | 0.001
$A_{T}^{1}$(fake) | - | 0.0019 | 0.0017 | 0.005 | 0.006
$A_{T}^{2}$(fake) | - | 0.0008 | 0.0008 | 0.003 | 0.003
$A_{T}^{3}$(fake) | - | 0.0015 | 0.0006 | 0.005 | 0.005
$A_{T}^{4}$(fake) | - | 0.0003 | 0.0004 | 0.007 | 0.007

##  8 Direct $C\\!P$ rate asymmetry

The raw measurement of the rate asymmetry is obtained from

$A=\frac{N({{\kern
1.79993pt\overline{\kern-1.79993ptB}{}}^{0}}\\!\rightarrow\phi{{\kern
1.99997pt\overline{\kern-1.99997ptK}{}}^{*0}})-N({{B}^{0}}\\!\rightarrow\phi{{K}^{*0}})}{N({{\kern
1.79993pt\overline{\kern-1.79993ptB}{}}^{0}}\\!\rightarrow\phi{{\kern
1.99997pt\overline{\kern-1.99997ptK}{}}^{*0}})+N({{B}^{0}}\\!\rightarrow\phi{{K}^{*0}})}\;.$
(15)

The numbers of events, $N$, are determined from fits to the $m_{KKK\pi}$
invariant mass distribution performed separately for ${B}^{0}$ and ${\kern
1.79993pt\overline{\kern-1.79993ptB}{}}^{0}$ decays, identified using the
charge of the final-state kaon. The dilution from the S-wave components is
corrected for using the results of the angular analysis.

The candidates are separated into the TIS and TOS trigger categories. In this
study, candidates that are accepted by both trigger decisions are included in
both categories and a possible bias to the central value is treated as a
systematic uncertainty. The obtained raw asymmetries for the two trigger types
are

$A^{\textrm{TOS}}_{\phi K^{*0}}=+0.014\pm 0.043\hskip
14.22636pt\textrm{and}\hskip 14.22636ptA^{\textrm{TIS}}_{\phi
K^{*0}}=-0.002\pm 0.040\;.$

The direct $C\\!P$ asymmetry is related to the measured $A$ by

$A_{{C\\!P}}=A-\delta\hskip 14.22636pt\textrm{with}\hskip
14.22636pt\delta=A_{\textrm{D}}+\kappa_{d}A_{\textrm{P}}\;,$ (16)

where $A_{\textrm{D}}$ is the detection asymmetry between $K^{+}\pi^{-}$ and
$K^{-}\pi^{+}$ final-states, $A_{\textrm{P}}$ is the asymmetry in production
rate between ${B}^{0}$ and ${\kern
1.79993pt\overline{\kern-1.79993ptB}{}}^{0}$ mesons in $pp$ collisions, and
the factor $\kappa_{d}$ accounts for the dilution of the production asymmetry
due to ${{B}^{0}}-{{\kern 1.79993pt\overline{\kern-1.79993ptB}{}}^{0}}$
oscillations.

The decay ${{B}^{0}}\rightarrow J/\psi K^{*0}$ is used as a control channel to
determine the difference in asymmetries

$\Delta A_{{C\\!P}}=A_{{C\\!P}}(\phi K^{*0})-A_{{C\\!P}}(J/\psi K^{*0})\;,$
(17)

since the detector and production asymmetries cancel in the difference.
Assuming $A_{CP}$ to be zero for the tree-level $B^{0}\rightarrow J/\psi
K^{*0}$ decay, $\Delta A_{{C\\!P}}$ is the $C\\!P$ asymmetry in
${{B}^{0}}\\!\rightarrow\phi{{K}^{*0}}$. The sample of ${{B}^{0}}\rightarrow
J/\psi K^{*0}$ decays, where the $J/\psi$ meson decays to a muon pair, are
collected through the same trigger and offline selections used for the signal
decay mode. Candidates are placed in the TOS trigger category if the trigger
decision is based on the decay products from the $K^{*0}$ meson only. Where
the decay products from the $J/\psi$ meson influences the trigger decision,
the candidate is rejected. The raw asymmetries obtained separately for the two
trigger types are

$A^{\textrm{TOS}}_{J/\psi K^{*0}}=-0.003\pm 0.016\hskip
14.22636pt\textrm{and}\hskip 14.22636ptA^{\textrm{TIS}}_{J/\psi
K^{*0}}=-0.016\pm 0.008\;.$

After averaging the trigger categories based on their statistical uncertainty,
the measured value for the difference in $C\\!P$ asymmetries is

$\Delta A_{{C\\!P}}=(+1.5\pm 3.2\pm 0.5)\,\%\;,$

where the uncertainties are statistical and systematic, respectively.
Systematic uncertainties arise from the differences between the event
topologies of the $B^{0}\rightarrow J/\psi K^{*0}$ and
${{B}^{0}}\\!\rightarrow\phi{{K}^{*0}}$ decays. Differences in the behaviour
of the events in the TIS trigger category between the signal and control modes
lead to an uncertainty of $0.25\,\%$. A further uncertainty of $0.4\,\%$
arises from the differences in kinematics of the daughter particles in the two
modes. The double counting of candidates in the overlap region leads to a
possible bias on the central value, estimated to be less than $0.1\,\%$.

##  9 Conclusions

In this paper measurements of the polarization amplitudes and strong phase
differences in the decay mode ${{B}^{0}}\\!\rightarrow\phi{{K}^{*0}}$ are
reported. The results for the P-wave parameters are shown in Table 5; these
are consistent with, but more precise than previous measurements. All
measurements are consistent with the presence of a large transverse component
rather than the naïve expectation of a dominant longitudinal polarization.

Table 5: Comparison of measurements made by the LHCb, BaBar [8] and Belle [11] collaborations. The first uncertainty is statistical and the second systematic.  Parameter | LHCb | BaBar | Belle
---|---|---|---
$f_{\textrm{L}}$ | $\phantom{-}0.497\pm 0.019\pm 0.015$ | $\phantom{-}0.494\pm 0.034\pm 0.013$ | $\phantom{-}0.499\pm 0.030\pm 0.018$
$f_{\perp}$ | $\phantom{-}0.221\pm 0.016\pm 0.013$ | $\phantom{-}0.212\pm 0.032\pm 0.013$ | $\phantom{-}0.238\pm 0.026\pm 0.008$
$\delta_{\perp}$ | $\phantom{-}2.633\pm 0.062\pm 0.037$ | $\phantom{-}2.35\phantom{0}\pm 0.13\phantom{0}\pm 0.09\phantom{0}$ | $\phantom{-}2.37\phantom{0}\pm 0.10\phantom{0}\pm 0.04\phantom{0}$
$\delta_{\parallel}$ | $\phantom{-}2.562\pm 0.069\pm 0.040$ | $\phantom{-}2.40\phantom{0}\pm 0.13\phantom{0}\pm 0.08\phantom{0}$ | $\phantom{-}2.23\phantom{0}\pm 0.10\phantom{0}\pm 0.02\phantom{0}$
$A_{0}^{{C\\!P}}$ | $-0.003\pm 0.038\pm 0.005$ | $+0.01\phantom{0}\pm 0.07\phantom{0}\pm 0.02\phantom{0}$ | $-0.030\pm 0.061\pm 0.007$
$A_{\perp}^{{C\\!P}}$ | $+0.047\pm 0.072\pm 0.009$ | $-0.04\phantom{0}\pm 0.15\phantom{0}\pm 0.06\phantom{0}$ | $-0.14\phantom{0}\pm 0.11\phantom{0}\pm 0.01\phantom{0}$
$\delta_{\perp}^{{C\\!P}}$ | $+0.062\pm 0.062\pm 0.006$ | $+0.21\phantom{0}\pm 0.13\phantom{0}\pm 0.08\phantom{0}$ | $+0.05\phantom{0}\pm 0.10\phantom{0}\pm 0.02\phantom{0}$
$\delta_{\parallel}^{{C\\!P}}$ | $+0.045\pm 0.068\pm 0.015$ | $+0.22\phantom{0}\pm 0.12\phantom{0}\pm 0.08\phantom{0}$ | $-0.02\phantom{0}\pm 0.10\phantom{0}\pm 0.01\phantom{0}$

It is more difficult to make comparisons for the S-wave components as this is
the first measurement to include consistently the effect of the S-wave in the
$K^{+}K^{-}$ system, and because the $K^{+}\pi^{-}$ mass range is different
with respect to the range used in previous analyses. The measurements of the
polarization amplitude differences are consistent with $C\\!P$ conservation.

The results of the angular analysis are used to determine triple-product
asymmetries. The measured true asymmetries show no evidence for $C\\!P$
violation. In contrast, large fake asymmetries are observed, indicating the
presence of significant final-state interactions. The difference in direct
$C\\!P$ asymmetries between the ${{B}^{0}}\\!\rightarrow\phi{{K}^{*0}}$ and
$B^{0}\rightarrow J/\psi K^{*0}$ decays is also measured,

$\displaystyle\Delta A_{{C\\!P}}=(+1.5\pm 3.2\pm 0.5)\,\%\;,$

where the first uncertainty is statistical and the second systematic. This is
a factor of two more precise than previous values reported by BaBar and Belle
[8, 11] and is found to be consistent with zero.

## 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); NASU (Ukraine); STFC (United Kingdom); NSF (USA). We also
acknowledge the support received from EPLANET and 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 indebted to the communities behind the multiple open source
software packages on which we depend. We are also thankful for the computing
resources and the access to software R&D tools provided by Yandex LLC
(Russia).

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