EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)

 CERN-PH-EP-2014-055 LHCb-PAPER-2014-006 23 Jun 2014

Differential branching fractions

and isospin asymmetries

of $B\\!\rightarrow K^{(*)}\mu^{+}\mu^{-}$ decays

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

The isospin asymmetries of $B\rightarrow K\mu^{+}\mu^{-}$ and $B\rightarrow
K^{*}\mu^{+}\mu^{-}$ decays and the partial branching fractions of the
$B^{0}\rightarrow K^{0}\mu^{+}\mu^{-}$, $B^{+}\rightarrow K^{+}\mu^{+}\mu^{-}$
and $B^{+}\rightarrow K^{*+}\mu^{+}\mu^{-}$ decays are measured as functions
of the dimuon mass squared, $q^{2}$. The data used correspond to an integrated
luminosity of 3 fb-1 from proton-proton collisions collected with the LHCb
detector at centre-of-mass energies of 7 TeV and 8 TeV in 2011 and 2012,
respectively. The isospin asymmetries are both consistent with the Standard
Model expectations. The three measured branching fractions favour lower values
than their respective theoretical predictions, however they are all
individually consistent with the Standard Model.

Submitted to JHEP

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

LHCb collaboration

R. Aaij41, 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, 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-Montells11, 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.
Fu21,38, E. Furfaro24,l, A. Gallas Torreira37, D. Galli14,d, S. Gallorini22,
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, 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,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. Nasteva2,
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

##  1 Introduction

The rare decay of a $B$ meson into a strange meson and a $\mu^{+}\mu^{-}$ pair
is a $b$ $\rightarrow$ $s$ quark-level transition. In the Standard Model (SM),
this can only proceed via loop diagrams. The loop-order suppression of the SM
amplitudes increases the sensitivity to new virtual particles that can
influence the decay amplitude at a similar level to the SM contribution. The
branching fractions of $B\\!\rightarrow K^{(*)}\mu^{+}\mu^{-}$ decays are
highly sensitive to contributions from vector or axial-vector like particles
predicted in extensions of the SM. However, despite recent progress in lattice
calculations [1, 2], theoretical predictions of the decay rates suffer from
relatively large uncertainties in the $B\rightarrow K^{(*)}$ form factor
calculations.

To maximise sensitivity, observables can be constructed from ratios or
asymmetries where the leading form factor uncertainties cancel. The
$C\\!P$-averaged isospin asymmetry ($A_{\rm I}$) is such an observable. It is
defined as

$\begin{split}A_{\rm I}&=\frac{\Gamma(B^{0}\\!\rightarrow
K^{(*)0}\mu^{+}\mu^{-})-\Gamma(B^{+}\\!\rightarrow
K^{(*)+}\mu^{+}\mu^{-})}{\Gamma(B^{0}\\!\rightarrow
K^{(*)0}\mu^{+}\mu^{-})+\Gamma(B^{+}\\!\rightarrow
K^{(*)+}\mu^{+}\mu^{-})}\phantom{A_{\rm I}}\phantom{,}\\\
&=\frac{\mathcal{B}(B^{0}\\!\rightarrow
K^{(*)0}\mu^{+}\mu^{-})-({\tau_{0}}/{\tau_{+}})\cdot\mathcal{B}(B^{+}\\!\rightarrow
K^{(*)+}\mu^{+}\mu^{-})}{\mathcal{B}(B^{0}\\!\rightarrow
K^{(*)0}\mu^{+}\mu^{-})+({\tau_{0}}/{\tau_{+}})\cdot\mathcal{B}(B^{+}\\!\rightarrow
K^{(*)+}\mu^{+}\mu^{-})},\end{split}$ (1)

where $\Gamma(f)$ and $\mathcal{B}(f)$ are the partial width and branching
fraction of the $B\rightarrow f$ decay and $\tau_{0}/\tau_{+}$ is the ratio of
the lifetimes of the $B^{0}$ and $B^{+}$ mesons111The inclusion of charge
conjugated processes is implied throughout this paper.. The decays in the
isospin ratio differ only by the charge of the light (spectator) quark in the
$B$ meson. The SM prediction for $A_{\rm I}$ is $\mathcal{O}(1\%)$ in the
dimuon mass squared, $q^{2}$, region below the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ resonance [3, 4, 5]. There is
no precise prediction for $A_{\rm I}$ for the $q^{2}$ region above the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ resonance, but it is expected
to be even smaller than at low $q^{2}$ [5]. As $q^{2}$ approaches zero, the
isospin asymmetry of $B\\!\rightarrow K^{*}\mu^{+}\mu^{-}$ is expected to
approach the same asymmetry as in $B\\!\rightarrow K^{*}\gamma$ decays, which
is measured to be $(5\pm 3)\%$ [6].

Previously, $A_{\rm I}$ has been measured by the BaBar [7], Belle [8] and LHCb
[9] collaborations, where measurements for the $B\\!\rightarrow
K\mu^{+}\mu^{-}$ decay have predominantly given negative values of $A_{\rm
I}$. In particular, the $B\\!\rightarrow K\mu^{+}\mu^{-}$ isospin asymmetry
measured by the LHCb experiment deviates from zero by over 4 standard
deviations. For $B\\!\rightarrow K^{*}\mu^{+}\mu^{-}$, measurements of $A_{\rm
I}$ are consistent with zero.

This paper describes a measurement of the isospin asymmetry in
$B\\!\rightarrow K\mu^{+}\mu^{-}$ and $B\\!\rightarrow K^{*}\mu^{+}\mu^{-}$
decays based on data collected with the LHCb detector, corresponding to an
integrated luminosity of 1$\mbox{\,fb}^{-1}$ recorded in 2011 at a centre-of-
mass energy $\sqrt{s}=7$ TeV, and 2$\mbox{\,fb}^{-1}$ recorded in 2012 at
$\sqrt{s}=8$ TeV. The previous analysis [9] was carried out on the
1$\mbox{\,fb}^{-1}$ of data recorded in 2011. The analysis presented here
includes, in addition to the data from 2012, a re-analysis of the full
1$\mbox{\,fb}^{-1}$ data sample with improved detector alignment parameters,
reconstruction algorithms and event selection. Thus it supersedes the
measurements in Ref. [9]. Moreover, the assumption that there is no isospin
asymmetry in the $B\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{(*)}$ decays is now used for all the measurements.

The isospin asymmetries are determined by measuring the differential branching
fractions of $B^{+}\\!\rightarrow K^{+}\mu^{+}\mu^{-}$, $B^{0}\\!\rightarrow
K^{0}\mu^{+}\mu^{-}$, $B^{0}\\!\rightarrow K^{*0}\mu^{+}\mu^{-}$ and
$B^{+}\\!\rightarrow K^{*+}\mu^{+}\mu^{-}$ decays. The $K^{0}$ meson is
reconstructed through the decay $K^{0}_{\rm\scriptscriptstyle
S}\rightarrow\pi^{+}\pi^{-}$; the $K^{*+}$ as $K^{*+}\rightarrow
K^{0}_{\rm\scriptscriptstyle S}(\rightarrow\pi^{+}\pi^{-})\pi^{+}$ and the
$K^{*0}$ as $K^{*0}\rightarrow K^{+}\pi^{-}$. Modes involving a
$K^{0}_{\rm\scriptscriptstyle L}$ or $\pi^{0}$ in the final state are not
considered. The individual branching fractions of $B^{+}\\!\rightarrow
K^{+}\mu^{+}\mu^{-}$, $B^{0}\\!\rightarrow K^{0}\mu^{+}\mu^{-}$ and
$B^{+}\\!\rightarrow K^{*+}\mu^{+}\mu^{-}$ decays are also reported. The
branching fraction of the decay $B^{0}\\!\rightarrow K^{*0}\mu^{+}\mu^{-}$ has
been previously reported in Ref. [10] and is not updated here.

The $B^{0}\\!\rightarrow K^{*0}\mu^{+}\mu^{-}$ and $B^{+}\\!\rightarrow
K^{*+}\mu^{+}\mu^{-}$ branching fractions are influenced by the presence of
$B^{0}\\!\rightarrow K^{+}\pi^{-}\mu^{+}\mu^{-}$ and $B^{+}\\!\rightarrow
K^{0}_{\rm\scriptscriptstyle S}\pi^{+}\mu^{+}\mu^{-}$ decays with the
$K^{+}\pi^{-}$ or $K^{0}_{\rm\scriptscriptstyle S}\pi^{+}$ system in a S-wave
configuration. It is not possible to separate these candidates from the
dominant $K^{*0}$ and $K^{*+}$ resonant components without performing an
analysis of the $K^{+}\pi^{-}$ or $K^{0}_{\rm\scriptscriptstyle S}\pi^{+}$
invariant mass and the angular distribution of the final state particles. The
S-wave component is expected to be at the level of a few percent [11] and to
cancel when evaluating the isospin asymmetry of the $B\\!\rightarrow
K^{*}\mu^{+}\mu^{-}$ decays.

##  2 Detector and dataset

The LHCb detector [12] 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 [13] placed downstream of the
magnet. 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 an impact parameter resolution
of 20${\,\upmu\rm m}$ for tracks with high transverse momentum. Charged
hadrons are identified using two ring-imaging Cherenkov (RICH) detectors [14].
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
[15]. Decays of $K^{0}_{\rm\scriptscriptstyle S}\\!\rightarrow\pi^{+}\pi^{-}$
are reconstructed in two different categories: the first involving
$K^{0}_{\rm\scriptscriptstyle S}$ mesons that decay early enough for the
daughter pions to be reconstructed in the vertex detector; and the second
containing $K^{0}_{\rm\scriptscriptstyle S}$ mesons that decay later such that
track segments of the pions cannot be formed in the vertex detector. These
categories are referred to as _long_ and _downstream_ , respectively.
Candidates in the long category have better mass, momentum and vertex
resolution than those in the downstream category.

Simulated events are used to estimate the efficiencies of the trigger,
reconstruction and subsequent event selection of the different signal decays
and to estimate the contribution from specific background sources. These
samples are produced using the software described in Refs. [16, 17, 18, 19,
20, 21, *Agostinelli:2002hh, 23].

##  3 Selection

The $B\\!\rightarrow K^{(*)}\mu^{+}\mu^{-}$ candidate events are required to
pass a two-stage trigger system [24]. In the initial hardware stage, these
events are selected with at least one muon with transverse momentum,
$\mbox{$p_{\rm T}$}>1.48\,(1.76){\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ in 2011
(2012). In the subsequent software stage, at least one of the final-state
particles is required to have $\mbox{$p_{\rm
T}$}>1.0{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ and an impact parameter (IP)
larger than $100{\,\upmu\rm m}$ with respect to all of the primary $pp$
interaction vertices (PVs) in the event. Finally, a multivariate algorithm
[25] is used for the identification of secondary vertices consistent with the
decay of a $b$ hadron with muons in the final state.

For the $B^{0}\\!\rightarrow K^{0}_{\rm\scriptscriptstyle S}\mu^{+}\mu^{-}$
and $B^{+}\\!\rightarrow K^{*+}\mu^{+}\mu^{-}$ modes,
$K^{0}_{\rm\scriptscriptstyle S}$ candidates are required to have a mass
within 30 ${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ of the known
$K^{0}_{\rm\scriptscriptstyle S}$ mass [26]. For the $B^{0}\\!\rightarrow
K^{*0}\mu^{+}\mu^{-}$ and $B^{+}\\!\rightarrow K^{*+}\mu^{+}\mu^{-}$ modes,
$K^{*}$ candidates are formed by combining kaons and pions and are required to
have a mass within 100${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ of the known
$K^{*}$ masses [26]. For all decay modes, $B$ candidates are formed by
subsequently combining the $K^{(*)}$ meson with two muons of opposite charge
and requiring the mass to be between 5170 and
5700${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$.

The event selection is common to that described in Refs. [27, 10, 28]: the
$\mu^{\pm}$ and the $K^{+}$ candidates are required to have $\chi^{2}_{\rm
IP}>9$, where $\chi^{2}_{\rm IP}$ is defined as the minimum change in
$\chi^{2}$ of the vertex fit to any of the PVs in the event when the particle
is added to that PV; the dimuon pair vertex fit has $\chi^{2}<9$; the $B$
candidate is required to have a vertex fit $\chi^{2}<8$ per degree of freedom;
the $B$ momentum vector is aligned with respect to one of the PVs in the event
within 14$\rm\,mrad$, the $B$ candidate has $\chi^{2}_{\rm IP}<9$ with respect
to that PV and the vertex fit $\chi^{2}$ of that PV increases by more than 121
when including the $B$ decay products. In addition, the
$K^{0}_{\rm\scriptscriptstyle S}$ candidate is required to have a decay time
larger than 2${\rm\,ps}$.

Using particle identification information from the RICH detectors,
calorimeters and muon system, multivariate discriminants (PID variables) are
employed to reject background candidates, where pions are misidentified as
kaons and vice-versa, and where a pion or kaon is incorrectly identified as a
muon.

The initial selection is followed by a tighter multivariate selection, based
on a boosted decision tree (BDT) [29] with the AdaBoost algorithm[30], which
is designed to reject background of combinatorial nature. Separate BDTs are
employed for each signal decay. For decays involving a
$K^{0}_{\rm\scriptscriptstyle S}$ meson, two independent BDTs are trained for
the long and downstream categories. This gives a total of six BDTs which all
use data from the upper mass sideband ($m(K^{(*)}\mu^{+}\mu^{-})>5350$
${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$) of their corresponding decay to
represent the background sample in the training. Simulated
$B^{+}\\!\rightarrow K^{+}\mu^{+}\mu^{-}$, $B^{0}\\!\rightarrow
K^{0}_{\rm\scriptscriptstyle S}\mu^{+}\mu^{-}$ and $B^{+}\\!\rightarrow
K^{*+}\mu^{+}\mu^{-}$ events are used as the signal sample in the training of
the corresponding BDTs. In contrast, to stay consistent with the selection in
Ref. [27], the signal for the training of the $B^{0}\\!\rightarrow
K^{*0}\mu^{+}\mu^{-}$ BDT is taken from reconstructed
$B^{0}\\!\rightarrow({J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\rightarrow\mu^{+}\mu^{-})K^{*0}$ candidates from data. Events used in
the training of the BDTs are not used in the subsequent classification of the
data.

All six BDTs use predominantly geometric variables, including the variables
used in the pre-selection described above. The $B^{0}\\!\rightarrow
K^{*0}\mu^{+}\mu^{-}$ BDT also makes use of PID variables to further suppress
background where a $K^{+}$ is misidentified as a $\pi^{+}$ and vice-versa in
the $K^{*0}$ decay.

The multivariate selections for $B^{+}\\!\rightarrow K^{+}\mu^{+}\mu^{-}$,
$B^{+}\\!\rightarrow K^{*+}\mu^{+}\mu^{-}$ and $B^{0}\\!\rightarrow
K^{*0}\mu^{+}\mu^{-}$ candidates have an efficiency of $90\%$ for the signal
channels and remove $95\%$ of the background that remains after the pre-
selection. The long lifetime of the $K^{0}_{\rm\scriptscriptstyle S}$ meson
makes it difficult to determine whether it originates from the same vertex as
the dimuon system in $B^{0}\\!\rightarrow K^{0}_{\rm\scriptscriptstyle
S}\mu^{+}\mu^{-}$ decays. As such, the multivariate selection for
$B^{0}\\!\rightarrow K^{0}_{\rm\scriptscriptstyle S}\mu^{+}\mu^{-}$ candidates
has a signal efficiency of 66% and 48% for the long and downstream categories,
respectively, while removing 99% of the background surviving the pre-
selection.

Combinatorial background, where the final-state particles attributed to the
$B$ candidate do not all come from the same $b$-hadron decay, are reduced to a
small level by the multivariate selection. In addition, there are several
sources of background that peak in the $K^{(*)}\mu^{+}\mu^{-}$ invariant mass.
The largest of these are $B\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{(*)}$ and $B\\!\rightarrow\psi{(2S)}K^{(*)}$ decays, which are
rejected by removing the regions of dimuon invariant mass around the
charmonium resonances
$(2828<m({\mu^{+}\mu^{-}})<3317{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ and
$3536<m({\mu^{+}\mu^{-}})<3873{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}})$. A
combination of mass and PID requirements remove additional peaking
backgrounds. These include $\mathchar
28931\relax^{0}_{b}\\!\rightarrow\mathchar 28931\relax^{(*)}\mu^{+}\mu^{-}$
decays, where the proton from the $\mathchar 28931\relax\\!\rightarrow
p\pi^{-}$ decay is misidentified as a $K^{+}$ or the proton misidentified as a
$\pi^{+}$ in the $\mathchar 28931\relax^{*}\\!\rightarrow pK^{-}$ decay,
$B^{0}_{s}\\!\rightarrow\phi\mu^{+}\mu^{-}$ decays where a kaon from
$\phi\\!\rightarrow K^{+}K^{-}$ is misidentified as a pion, and
$B^{+}\\!\rightarrow K^{+}\mu^{+}\mu^{-}$ decays that combine with a random
pion to fake a $B^{0}\\!\rightarrow K^{*0}\mu^{+}\mu^{-}$ decay. After the
application of all the selection criteria the exclusive backgrounds are
reduced to less than $1\%$ of the level of the signal.

To improve the resolution on the reconstructed mass of the $B$ meson, a
kinematic fit [31] is performed for candidates involving a
$K^{0}_{\rm\scriptscriptstyle S}$ meson. In the fit, the mass of the
$\pi^{+}\pi^{-}$ system is constrained to the nominal
$K^{0}_{\rm\scriptscriptstyle S}$ mass and the $B$ candidate is required to
originate from its associated PV.

##  4 Signal yield determination

Signal yields are determined using extended unbinned maximum likelihood fits
to the $K^{(*)}\mu^{+}\mu^{-}$ mass in the range
5170–5700${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$. These fits are performed
in nine bins of $q^{2}$ for $B^{0}\\!\rightarrow K^{0}_{\rm\scriptscriptstyle
S}\mu^{+}\mu^{-}$, $B^{+}\\!\rightarrow K^{*+}\mu^{+}\mu^{-}$ and
$B^{0}\\!\rightarrow K^{*0}\mu^{+}\mu^{-}$ decays, while for the
$B^{+}\\!\rightarrow K^{+}\mu^{+}\mu^{-}$ decay the larger number of signal
events allows to define nineteen $q^{2}$ bins. The binning scheme is shown in
Tables 4 to 6 of the appendix. It removes the region of $q^{2}$ around the
charmonium resonances. For the $B^{+}\\!\rightarrow K^{+}\mu^{+}\mu^{-}$
differential branching fraction, where the statistical uncertainty is the
smallest, a narrow range in $m(\mu^{+}\mu^{-})$ is also removed around the
mass of the $\phi$ meson. The signal component in the fit is described by the
sum of two Crystal Ball functions [32] with common peak values and tail
parameters, but different widths. The signal shape parameters are taken from a
fit to $B\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{(*)}$
channels in the data, with a correction that accounts for a small $q^{2}$
dependence on the peak value and width obtained from the simulation. The
combinatorial background is parameterised by an exponential function, which is
allowed to vary for each $q^{2}$ bin and $K^{0}_{\rm\scriptscriptstyle S}$
category independently. For decays involving $K^{0}_{\rm\scriptscriptstyle S}$
mesons, separate fits are made to the long and downstream categories. The mass
fits for the four signal channels are shown in Fig. 1, where the long and
downstream $K^{0}_{\rm\scriptscriptstyle S}$ categories are combined and the
results of the fits, performed in separate $q^{2}$ bins, are merged for
presentation purposes. The corresponding number of signal candidates for each
channel is given in Table 1.

Table 1: Observed yields of the four signal channels summed over the $q^{2}$ bins, excluding the charmonium resonance regions. Only the statistical uncertainties are shown. Decay mode | Signal yield
---|---
$B^{+}\\!\rightarrow K^{+}\mu^{+}\mu^{-}$ | $4746\pm 81$
$B^{0}\\!\rightarrow K^{0}_{\rm\scriptscriptstyle S}\mu^{+}\mu^{-}$ | $176\pm 17$
$B^{+}\\!\rightarrow K^{*+}(\rightarrow K^{0}_{\rm\scriptscriptstyle S}\pi^{+})\mu^{+}\mu^{-}$ | $162\pm 16$
$B^{0}\\!\rightarrow K^{*0}(\rightarrow K^{+}\pi^{-})\mu^{+}\mu^{-}$ | $2361\pm 56$

Figure 1: Reconstructed $B$ candidate mass for the four signal modes. The data
are overlaid with the result of the fit described in the text. The long and
downstream $K^{0}_{\rm\scriptscriptstyle S}$ categories are combined. The
results of the fits, performed in separate $q^{2}$ bins, are merged for
presentation purposes. The blue (shaded) region is the combinatorial
background.

##  5 Branching fraction normalisation

Each signal mode is normalised with respect to its corresponding
$B\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{(*)}$ channel,
where the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ resonance decays into
two muons. These normalisation channels have branching fractions that are
approximately two orders of magnitude higher than those of the signal
channels. Each normalisation channel has similar kinematic properties and the
same final-state particles as the signal modes. This results in an almost
complete cancellation of systematic uncertainties when measuring the ratio of
branching fractions of the signal mode with the corresponding normalisation
channel. Separate normalisations for the long and downstream
$K^{0}_{\rm\scriptscriptstyle S}$ reconstruction categories are used to
further cancel potential sources of systematic uncertainty.

Corrections to the IP resolution, PID variables and $B$ candidate kinematic
properties are applied to the simulated events, such that the distributions of
simulated candidates from the normalisation channels agree with the data. The
simulation samples are subsequently used to calculate the relative
efficiencies as functions of $q^{2}$. The $q^{2}$ dependence arises mainly
from trigger effects, where the muons have increased (decreased) $p_{\rm T}$
at high (low) $q^{2}$ and consequently have a higher (lower) trigger
efficiency. Furthermore, at high $q^{2}$, the hadrons are almost at rest in
the $B$ meson rest frame and, like the $B$ meson, points back to the PV in the
laboratory frame. The IP requirements applied on the hadron have a lower
efficiency for this region of $q^{2}$. The $K^{0}_{\rm\scriptscriptstyle S}$
channels have an additional effect due to the different acceptance of the two
reconstruction categories; $K^{0}_{\rm\scriptscriptstyle S}$ mesons are more
likely to be reconstructed in the long category if they have low momentum,
which favours the high $q^{2}$ region. The momentum distributions of the
$K^{0}_{\rm\scriptscriptstyle S}$ mesons in
$B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{0}_{\rm\scriptscriptstyle S}$ and
$B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{*+}$ decays
in data and simulation for both $K^{0}_{\rm\scriptscriptstyle S}$ categories
are in good agreement, indicating that the acceptance is well described in the
simulation.

The measured differential branching fraction averaged over a $q^{2}$ bin of
width $q^{2}_{\rm max}-q^{2}_{\rm min}$ is given by

$\frac{d\mathcal{B}}{dq^{2}}=\frac{N(B\\!\rightarrow
K^{(*)}\mu^{+}\mu^{-})}{N(B\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{(*)})}\cdot\frac{\varepsilon(B\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{(*)})}{\varepsilon(B\\!\rightarrow
K^{(*)}\mu^{+}\mu^{-})}\cdot\frac{\mathcal{B}(B\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{(*)})\mathcal{B}({J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\rightarrow\mu^{+}\mu^{-})}{(q^{2}_{\rm max}-q^{2}_{\rm min})},$ (2)

where ${N(B\\!\rightarrow K^{(*)}\mu^{+}\mu^{-}})$ is the number of signal
candidates in the bin,
${N(B\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{(*)}})$ is
the number of normalisation candidates, the product of
$\mathcal{B}(B\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{(*)})$ and $\mathcal{B}({J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\rightarrow\mu^{+}\mu^{-})$ is the visible branching fraction of the
normalisation channel, and ${\varepsilon(B\\!\rightarrow
K^{(*)}\mu^{+}\mu^{-})}/{\varepsilon(B\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{(*)})}$ is the relative efficiency between the signal and
normalisation channels in the bin.

##  6 Systematic uncertainties

The branching fraction measurements of the normalisation modes from the
$B$-factory experiments assume that the $B^{+}$ and $B^{0}$ mesons are
produced with equal proportions at the $\Upsilon(4S)$ resonance [33, 34, 35].
In contrast, in this paper isospin symmetry is assumed for the
$B\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{(*)}$ decays,
implying that the $B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{+}$ ($B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{*+}$) and $B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{0}$ ($B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{*0}$) decays have the same partial width. The branching fractions
used in the normalisation are obtained by: taking the most precise branching
fraction results from Ref. [33] and translating them into partial widths;
averaging the partial widths of the $K^{+}$, $K^{0}$ and the $K^{*+}$,
$K^{*0}$ modes, respectively; and finally translating the widths back to
branching fractions. The calculation only requires knowledge of the ratio of
$B^{0}$ and $B^{+}$ lifetimes for which we use $0.93\pm 0.01$ [26].
Statistical uncertainties are treated as uncorrelated while systematical
uncertainties are conservatively treated as fully correlated. The resulting
branching fractions of the normalisation channels are

$\begin{split}\mathcal{B}(B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{+})&=(0.998\pm 0.014\pm 0.040)\times 10^{-3},\\\
\mathcal{B}(B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{0})&=(0.928\pm 0.013\pm 0.037)\times 10^{-3},\\\
\mathcal{B}(B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{*+})&=(1.431\pm 0.027\pm 0.090)\times 10^{-3},\\\
\mathcal{B}(B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{*0})&=(1.331\pm 0.025\pm 0.084)\times 10^{-3},\end{split}$

where the first uncertainty is statistical and the second systematic.

A systematic uncertainty is assigned to account for the imperfect knowledge of
the $q^{2}$ spectrum in the simulation within each $q^{2}$ bin. For example,
the recent observation of a resonance in the high $q^{2}$ region of
$B^{+}\\!\rightarrow K^{+}\mu^{+}\mu^{-}$ decays [27] alters the $q^{2}$
distribution and hence the selection efficiencies in that region. By
reweighting simulated events to account for this resonance, and for variations
of the $B\rightarrow K^{(*)}$ form factor model as described in Ref. [36], a
systematic uncertainty is determined at the level of $(1-2)\%$ depending on
channel and $q^{2}$ bin.

Data-driven corrections of the long and downstream tracking efficiencies in
the simulation are determined using tag-and-probe techniques in
$J/\psi\rightarrow\mu^{+}\mu^{-}$ and $D^{0}\rightarrow\phi
K^{0}_{\rm\scriptscriptstyle S}$ decays, respectively. For the
$J/\psi\rightarrow\mu^{+}\mu^{-}$ decay, the tag is a fully reconstructed muon
track. It is combined with another muon, referred to as the probe,
reconstructed using the muon stations and the large-area silicon detector
upstream of the magnet. The tracking efficiency is determined by
reconstructing the probe using the full tracking system. The
$D^{0}\rightarrow\phi K^{0}_{\rm\scriptscriptstyle S}$ decay is tagged via a
partial reconstruction using only one of the $K^{0}_{\rm\scriptscriptstyle S}$
daughters. The downstream tracking efficiency is then evaluated by fully
reconstructing the $K^{0}_{\rm\scriptscriptstyle S}$ candidate. The resulting
systematic uncertainty on the efficiency ratio, due to finite precision of the
measurement, is found to be negligible. The systematic uncertainty that arises
from the corrections to the IP resolution, PID variables and $B$ candidate
kinematic properties in the simulation varies between 1% and 3% depending on
channel and $q^{2}$ bin.

A summary of the systematic uncertainties can be found in Table 2. The
uncertainties on the branching fractions of the normalisation modes constitute
the dominant source of systematic uncertainty on the branching fraction
measurements while it cancels in the isospin measurements.

Table 2: Summary of systematic uncertainties associated with the branching fraction and isospin asymmetry measurements. Source | Branching fraction | Isospin asymmetry
---|---|---
$B\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{(*)}$ branching fractions |  $4\%-6$% | $-$
Physics model |  $1\%-2$% |  $1\%-2$%
Simulation mis-modelling |  $1\%-3$% |  $1\%-3$%

##  7 Branching fraction results

The differential branching fraction results for $B^{+}\\!\rightarrow
K^{+}\mu^{+}\mu^{-}$, $B^{0}\\!\rightarrow K^{0}\mu^{+}\mu^{-}$ and
$B^{+}\\!\rightarrow K^{*+}\mu^{+}\mu^{-}$ decays are shown in Fig. 2 with
theoretical predictions [37, 38] superimposed. The values are given in Tables
4 to 6 in the appendix. In the low $q^{2}$ region, these predictions rely on
the QCD factorisation approaches from Refs. [39, 40] for $B\\!\rightarrow
K^{*}\mu^{+}\mu^{-}$ and Ref. [41] for $B\\!\rightarrow K\mu^{+}\mu^{-}$, and
lose accuracy when approaching the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}$ resonance. In the high $q^{2}$ region, an operator product expansion
in the inverse $b$-quark mass, $1/m_{b}$, and in $1/\sqrt{q^{2}}$ is used
based on Ref. [42]. This expansion is only valid above the open charm
threshold. A dimensional estimate of the uncertainty associated with this
expansion is discussed in Ref. [43]. For light cone sum rule (LCSR)
predictions, the $B\rightarrow K^{(*)}$ form factor calculations are taken
from Refs. [44] and [45]. Predictions based on form factors from lattice
calculations are also overlaid [1, 46, 2, 47].

Figure 2: Differential branching fraction results for the $B^{+}\\!\rightarrow
K^{+}\mu^{+}\mu^{-}$, $B^{0}\\!\rightarrow K^{0}\mu^{+}\mu^{-}$ and
$B^{+}\\!\rightarrow K^{*+}\mu^{+}\mu^{-}$ decays. The uncertainties shown on
the data points are the quadratic sum of the statistical and systematic
uncertainties. The shaded regions illustrate the theoretical predictions and
their uncertainties from light cone sum rule and lattice QCD calculations.

Although all three differential branching fraction measurements are consistent
with the SM, they all have values smaller than the theoretical prediction. The
sample size for $B^{+}\\!\rightarrow K^{+}\mu^{+}\mu^{-}$ is sufficient to
show significant structures in the $q^{2}$ distribution. As an example, the
peak at high $q^{2}$ is due to the $\psi(4160)$ resonance, which is discussed
in more detail in Ref. [27].

The presence of an S-wave contribution to the $K^{+}\pi^{-}$ and
$K^{0}_{\rm\scriptscriptstyle S}\pi^{+}$ systems of $B^{0}\\!\rightarrow
K^{*0}\mu^{+}\mu^{-}$ and $B^{+}\\!\rightarrow K^{*+}\mu^{+}\mu^{-}$
candidates, respectively, complicates the analysis of these channels. This
effect is of the order of a few percent and can be neglected in
$B^{+}\\!\rightarrow K^{*+}\mu^{+}\mu^{-}$ decays with the current statistical
precision. The larger signal yield of $B^{0}\\!\rightarrow
K^{*0}\mu^{+}\mu^{-}$, however, merits a detailed analysis of the S-wave
contribution and requires a dedicated study. For this reason the branching
fraction of $B^{0}\\!\rightarrow K^{*0}\mu^{+}\mu^{-}$ decays is not reported.

By convention, branching fractions are extrapolated to the full $q^{2}$ range
ignoring the presence of the narrow charmonium resonances. A $q^{2}$
distribution based on Ref. [48] is used for this. The correction factors to
the branching fractions due to this extrapolation are 1.39 and 1.50 for
$B\\!\rightarrow K\mu^{+}\mu^{-}$ and $B^{0}\\!\rightarrow
K^{*0}\mu^{+}\mu^{-}$, respectively. No uncertainty is assigned to these
corrections. Summing the $q^{2}$ bins and applying the extrapolation, the
integrated branching fractions become

$\begin{split}\mathcal{B}(B^{+}\\!\rightarrow K^{+}\mu^{+}\mu^{-})&=(4.29\pm
0.07\,(\rm{stat})\pm 0.21\,(\rm{syst}))\times 10^{-7},\phantom{,}\\\
\mathcal{B}(B^{0}\\!\rightarrow K^{0}\mu^{+}\mu^{-})&=(3.27\pm
0.34\,(\rm{stat})\pm 0.17\,(\rm{syst}))\times 10^{-7},\phantom{,}\\\
\mathcal{B}(B^{+}\\!\rightarrow K^{*+}\mu^{+}\mu^{-})&=(9.24\pm
0.93\,(\rm{stat})\pm 0.67\,(\rm{syst}))\times
10^{-7}.\phantom{and}\end{split}$

These measurements are more precise than the current world averages [26].

Table 3 compares the $B^{+}\\!\rightarrow K^{+}\mu^{+}\mu^{-}$ and
$B^{0}\\!\rightarrow K^{0}\mu^{+}\mu^{-}$ branching fractions integrated over
the $q^{2}$ region of $15-22{\mathrm{\,Ge\kern-1.00006ptV^{2}\\!/}c^{4}}$, and
the $B^{+}\\!\rightarrow K^{*+}\mu^{+}\mu^{-}$ branching fraction integrated
over the $15-19{\mathrm{\,Ge\kern-1.00006ptV^{2}\\!/}c^{4}}$ region to the
lattice QCD predictions [1, 46, 2, 47]. While the measurements are all
individually consistent with their respective predictions, they all have
values below those.

Table 3: Integrated branching fractions $(10^{-8})$ in the high $q^{2}$ region. For the $B\\!\rightarrow K\mu^{+}\mu^{-}$ modes the region is defined as $15-22{\mathrm{\,Ge\kern-0.90005ptV^{2}\\!/}c^{4}}$, while for $B^{+}\\!\rightarrow K^{*+}\mu^{+}\mu^{-}$ it is $15-19{\mathrm{\,Ge\kern-0.90005ptV^{2}\\!/}c^{4}}$. Predictions are obtained using the form factors calculated in lattice QCD over the same $q^{2}$ regions. For the measurements, the first uncertainty is statistical and the second systematic. Decay mode | Measurement | Prediction
---|---|---
$B^{+}\\!\rightarrow K^{+}\mu^{+}\mu^{-}$ | $\phantom{0}8.5\pm 0.3\pm 0.4$ | $10.7\pm 1.2$
$B^{0}\\!\rightarrow K^{0}\mu^{+}\mu^{-}$ | $\phantom{0}6.7\pm 1.1\pm 0.4$ | $\phantom{0}9.8\pm 1.0$
$B^{+}\\!\rightarrow K^{*+}\mu^{+}\mu^{-}$ | $15.8~{}^{+3.2}_{-2.9}\pm 1.1$ | $26.8\pm 3.6$

##  8 Isospin asymmetry results

The assumption of no isospin asymmetry in the
$B\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{(*)}$ modes
makes the isospin measurement equivalent to measuring the difference in
isospin asymmetry between $B\\!\rightarrow K^{(*)}\mu^{+}\mu^{-}$ and
$B\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{(*)}$ decays.
Compared to using the values in Ref. [26] for the branching fractions of the
$B\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{(*)}$ modes,
this approach shifts $A_{\rm I}$ in each bin by approximately 4%. The isospin
asymmetries are shown in Fig. 3 for $B\\!\rightarrow K\mu^{+}\mu^{-}$ and
$B\\!\rightarrow K^{*}\mu^{+}\mu^{-}$ and given in Tables 7 and 8 in the
appendix. The asymmetric uncertainties are obtained from the profile
likelihood.

Since there is no knowledge on the shape of $A_{\rm I}$ in models that extend
the SM, apart from large correlations expected between neighbouring bins, the
$A_{\rm I}=0$ hypothesis is tested against the simplest alternative, that is a
constant value different from zero. The difference in $\chi^{2}$ between the
two hypotheses is used as a test statistic and is compared to the differences
in an ensemble of pseudo-experiments which are generated with zero isospin
asymmetry. Given the current statistical precision, the hypothesis of $A_{\rm
I}=0$ is a good approximation to the SM which predicts $A_{\rm I}$ to be
$\mathcal{O}(1\%)$ [3, 4, 5]. The $p$-value for the $B\\!\rightarrow
K\mu^{+}\mu^{-}$ isospin asymmetry under the $A_{\rm I}=0$ hypothesis is 11%,
corresponding to a significance of 1.5 $\sigma$. The $B\\!\rightarrow
K^{*}\mu^{+}\mu^{-}$ isospin asymmetry has a $p$-value of 80% with respect to
zero. Alternatively, a simple $\chi^{2}$ test of the data with respect to a
hypothesis of zero isospin asymmetry has a $p$-value of 54% (4%) for the
$B\\!\rightarrow K\mu^{+}\mu^{-}$ ($B\\!\rightarrow K^{*}\mu^{+}\mu^{-}$)
isospin asymmetry.

Figure 3: Isospin asymmetries for (left) $B\\!\rightarrow K\mu^{+}\mu^{-}$ and
(right) $B\\!\rightarrow K^{*}\mu^{+}\mu^{-}$ decays. Figure 4: Isospin
asymmetry of $B\\!\rightarrow K\mu^{+}\mu^{-}$ obtained separately from the
2011 and 2012 data sets.

Although the isospin asymmetry for $B\\!\rightarrow K\mu^{+}\mu^{-}$ decays is
negative in all but one $q^{2}$ bin, results are more consistent with the SM
compared to the previous measurement in Ref. [9], which quoted a 4.4 $\sigma$
significance to differ from zero, using a test statistic that explicitly
tested for $A_{\rm I}$ to be negative in all bins. The lower significance
quoted here is due to four effects: the change of the test statistic in the
calculation of the significance itself, which reduces the previous discrepancy
to 3.5 $\sigma$; the assumption that the isospin asymmetry of
$B\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{(*)}$ is zero
which reduces the significance further to 3.2 $\sigma$; a re-analysis of the
2011 data with the updated reconstruction and event selection that reduces the
significance to 2.5 $\sigma$; and finally the inclusion of the 2012 data set
reduces the significance further to 1.5 $\sigma$.

The measurements of $A_{\rm I}$ in the individual $q^{2}$ bins obtained from
the re-analysis of the 2011 data set are compatible with those obtained in the
previous analysis; a $\chi^{2}$ test on the compatibility of the two results,
taking the overlap of events into account, has a $p$-value of 93%. However
results from the 2012 data are more compatible with an $A_{\rm I}$ value of
zero than the re-analysed 2011 data, as shown in Fig 4.

##  9 Conclusion

The most precise measurements of the differential branching fractions of
$B^{+}\\!\rightarrow K^{+}\mu^{+}\mu^{-}$, $B^{0}\\!\rightarrow
K^{0}_{\rm\scriptscriptstyle S}\mu^{+}\mu^{-}$ and $B^{+}\\!\rightarrow
K^{*+}\mu^{+}\mu^{-}$ decays as well as the isospin asymmetries of
$B\\!\rightarrow K\mu^{+}\mu^{-}$ and $B\\!\rightarrow K^{*}\mu^{+}\mu^{-}$
decays have been performed using a data set corresponding to
3$\mbox{\,fb}^{-1}$ of integrated luminosity collected by the LHCb detector.

The isospin asymmetries of the $B\\!\rightarrow K\mu^{+}\mu^{-}$ and
$B\\!\rightarrow K^{*}\mu^{+}\mu^{-}$ decays are both consistent with SM
expectations. However, the branching fraction measurements all have lower
values than the SM predictions. This is consistent with the
$B^{0}\\!\rightarrow K^{*0}\mu^{+}\mu^{-}$ and
$B^{0}_{s}\rightarrow\phi\mu^{+}\mu^{-}$ branching fractions measured by LHCb,
which also favour lower values [10, 49, 47] than predicted by the SM.

## Acknowledgements

We would like to thank Roman Zwicky for the useful discusions regarding
$A_{\rm I}$ in the $B\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{(*)}$ system, and Chris Bouchard and Stefan Meinel for information on
branching fraction predictions of $B\\!\rightarrow K\mu^{+}\mu^{-}$ and
$B\\!\rightarrow K^{*}\mu^{+}\mu^{-}$ from the lattice calculations. 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 and the Royal Society (United Kingdom);
NSF (USA). We also acknowledge the support received from EPLANET, Marie Curie
Actions 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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##  Appendix A Appendix

Table 4: Differential branching fraction results ($10^{-9}\times c^{4}/\mbox{GeV}^{2}$) for the $B^{+}\\!\rightarrow K^{+}\mu^{+}\mu^{-}$ decay, including statistical and systematic uncertainties. $q^{2}$ range ($\mathrm{\,Ge\kern-1.00006ptV}^{2}/c^{4}$) | central value | stat | syst
---|---|---|---
0.1 $<$ $q^{2}$ $<$ 0.98 | 33.2 | 1.8 | 1.7
1.1 $<$ $q^{2}$ $<$ 2.0 | 23.3 | 1.5 | 1.2
2.0 $<$ $q^{2}$ $<$ 3.0 | 28.2 | 1.6 | 1.4
3.0 $<$ $q^{2}$ $<$ 4.0 | 25.4 | 1.5 | 1.3
4.0 $<$ $q^{2}$ $<$ 5.0 | 22.1 | 1.4 | 1.1
5.0 $<$ $q^{2}$ $<$ 6.0 | 23.1 | 1.4 | 1.2
6.0 $<$ $q^{2}$ $<$ 7.0 | 24.5 | 1.4 | 1.2
7.0 $<$ $q^{2}$ $<$ 8.0 | 23.1 | 1.4 | 1.2
11.0 $<$ $q^{2}$ $<$ 11.8 | 17.7 | 1.3 | 0.9
11.8 $<$ $q^{2}$ $<$ 12.5 | 19.3 | 1.2 | 1.0
15.0 $<$ $q^{2}$ $<$ 16.0 | 16.1 | 1.0 | 0.8
16.0 $<$ $q^{2}$ $<$ 17.0 | 16.4 | 1.0 | 0.8
17.0 $<$ $q^{2}$ $<$ 18.0 | 20.6 | 1.1 | 1.0
18.0 $<$ $q^{2}$ $<$ 19.0 | 13.7 | 1.0 | 0.7
19.0 $<$ $q^{2}$ $<$ 20.0 |  07.4 | 0.8 | 0.4
20.0 $<$ $q^{2}$ $<$ 21.0 |  05.9 | 0.7 | 0.3
21.0 $<$ $q^{2}$ $<$ 22.0 |  04.3 | 0.7 | 0.2
1.1 $<$ $q^{2}$ $<$ 6.0 | 24.2 | 0.7 | 1.2
15.0 $<$ $q^{2}$ $<$ 22.0 | 12.1 | 0.4 | 0.6
Table 5: Differential branching fraction results ($10^{-9}\times c^{4}/\mbox{GeV}^{2}$) for the $B^{0}\\!\rightarrow K^{0}\mu^{+}\mu^{-}$ decay, including statistical and systematic uncertainties. $q^{2}$ range ($\mathrm{\,Ge\kern-1.00006ptV}^{2}/c^{4}$) | central value | stat | syst
---|---|---|---
0.1 $<$ $q^{2}$ $<$ 2.0 | 12.2 | ${}^{+5.9}_{-5.2}$ | 0.6
2.0 $<$ $q^{2}$ $<$ 4.0 | 18.7 | ${}^{+5.5}_{-4.9}$ | 0.9
4.0 $<$ $q^{2}$ $<$ 6.0 | 17.3 | ${}^{+5.3}_{-4.8}$ | 0.9
6.0 $<$ $q^{2}$ $<$ 8.0 | 27.0 | ${}^{+5.8}_{-5.3}$ | 1.4
11.0 $<$ $q^{2}$ $<$ 12.5 | 12.7 | ${}^{+4.5}_{-4.0}$ | 0.6
15.0 $<$ $q^{2}$ $<$ 17.0 | 14.3 | ${}^{+3.5}_{-3.2}$ | 0.7
17.0 $<$ $q^{2}$ $<$ 22.0 |  07.8 | ${}^{+1.7}_{-1.5}$ | 0.4
1.1 $<$ $q^{2}$ $<$ 6.0 | 18.7 | ${}^{+3.5}_{-3.2}$ | 0.9
15.0 $<$ $q^{2}$ $<$ 22.0 |  09.5 | ${}^{+1.6}_{-1.5}$ | 0.5
Table 6: Differential branching fraction results ($10^{-9}\times c^{4}/\mbox{GeV}^{2}$) for the $B^{+}\\!\rightarrow K^{*+}\mu^{+}\mu^{-}$ decay, including statistical and systematic uncertainties. $q^{2}$ range ($\mathrm{\,Ge\kern-1.00006ptV}^{2}/c^{4}$) | central value | stat | syst
---|---|---|---
0.1 $<$ $q^{2}$ $<$ 2.0 | 59.2 | ${}^{+14.4}_{-13.0}$ | 4.0
2.0 $<$ $q^{2}$ $<$ 4.0 | 55.9 | ${}^{+15.9}_{-14.4}$ | 3.8
4.0 $<$ $q^{2}$ $<$ 6.0 | 24.9 | ${}^{+11.0}_{-\phantom{0}9.6}$ | 1.7
6.0 $<$ $q^{2}$ $<$ 8.0 | 33.0 | ${}^{+11.3}_{-\phantom{0}10.0}$ | 2.3
11.0 $<$ $q^{2}$ $<$ 12.5 | 82.8 | ${}^{+15.8}_{-14.1}$ | 5.6
15.0 $<$ $q^{2}$ $<$ 17.0 | 64.4 | ${}^{+12.9}_{-11.5}$ | 4.4
17.0 $<$ $q^{2}$ $<$ 19.0 | 11.6 | ${}^{+\phantom{0}9.1}_{-\phantom{0}7.6}$ | 0.8
1.1 $<$ $q^{2}$ $<$ 6.0 | 36.6 | ${}^{+\phantom{0}8.3}_{-\phantom{0}7.6}$ | 2.6
15 $<$ $q^{2}$ $<$ 19.0 | 39.5 | ${}^{+\phantom{0}8.0}_{-\phantom{0}7.3}$ | 2.8
Table 7: Isospin asymmetry results for the $B\\!\rightarrow K\mu^{+}\mu^{-}$ decay, including statistical and systematic uncertainties. $q^{2}$ range ($\mathrm{\,Ge\kern-1.00006ptV}^{2}/c^{4}$) | central value | stat | syst
---|---|---|---
0.1 $<$ $q^{2}$ $<$ 2.0 | -0.37 | ${}^{+0.18}_{-0.21}$ | 0.02
2.0 $<$ $q^{2}$ $<$ 4.0 | -0.15 | ${}^{+0.13}_{-0.15}$ | 0.02
4.0 $<$ $q^{2}$ $<$ 6.0 | -0.10 | ${}^{+0.13}_{-0.16}$ | 0.02
6.0 $<$ $q^{2}$ $<$ 8.0 |  -0.09 | ${}^{+0.10}_{-0.11}$ | 0.02
11.0 $<$ $q^{2}$ $<$ 12.5 | -0.16 | ${}^{+0.15}_{-0.18}$ | 0.03
15.0 $<$ $q^{2}$ $<$ 17.0 | -0.04 | ${}^{+0.11}_{-0.13}$ | 0.02
17.0 $<$ $q^{2}$ $<$ 22.0 | -0.12 | ${}^{+0.10}_{-0.11}$ | 0.02
1.1 $<$ $q^{2}$ $<$ 6.0 | -0.10 | ${}^{+0.08}_{-0.09}$ | 0.02
15.0 $<$ $q^{2}$ $<$ 22.0 | -0.09 | ${}^{+0.08}_{-0.08}$ | 0.02
Table 8: Isospin asymmetry results for the $B\\!\rightarrow K^{*}\mu^{+}\mu^{-}$ decay, including statistical and systematic uncertainties. $q^{2}$ range ($\mathrm{\,Ge\kern-1.00006ptV}^{2}/c^{4}$) | central value | stat | syst
---|---|---|---
0.1 $<$ $q^{2}$ $<$ 2.0 |  -0.11 | ${}^{+0.12}_{-0.11}$ | 0.02
2.0 $<$ $q^{2}$ $<$ 4.0 | -0.20 | ${}^{+0.15}_{-0.12}$ | 0.03
4.0 $<$ $q^{2}$ $<$ 6.0 |  -0.23 | ${}^{+0.21}_{-0.18}$ | 0.02
6.0 $<$ $q^{2}$ $<$ 8.0 |  -0.19 | ${}^{+0.17}_{-0.15}$ | 0.02
11.0 $<$ $q^{2}$ $<$ 12.5 | -0.25 | ${}^{+0.09}_{-0.08}$ | 0.03
15.0 $<$ $q^{2}$ $<$ 17.0 | -0.10 | ${}^{+0.10}_{-0.09}$ | 0.03
17.0 $<$ $q^{2}$ $<$ 19.0 |  -0.51 | ${}^{+0.29}_{-0.24}$ | 0.02
1.1 $<$ $q^{2}$ $<$ 6.0 |  -0.00 | ${}^{+0.12}_{-0.10}$ | 0.02
15.0 $<$ $q^{2}$ $<$ 19.0 |  -0.06 | ${}^{+0.10}_{-0.09}$ | 0.02