EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)

 CERN-PH-EP-2013-070 LHCb-PAPER-2013-022 June 4, 2013

Measurement of the branching fractions of the decays
$B^{0}_{s}\rightarrow\kern
4.14793pt\overline{\kern-4.14793ptD}{}^{0}K^{-}\pi^{+}$ and
$B^{0}\rightarrow\kern 4.14793pt\overline{\kern-4.14793ptD}{}^{0}K^{+}\pi^{-}$

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

The first observation of the decay $B^{0}_{s}\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}K^{-}\pi^{+}$ is reported. The
analysis is based on a data sample, corresponding to an integrated luminosity
of $1.0\mbox{\,fb}^{-1}$ of $pp$ collisions, collected with the LHCb detector.
The branching fraction relative to that of the topologically similar decay
$B^{0}\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}\pi^{+}\pi^{-}$ is measured to be

$\frac{{\cal B}\left(B^{0}_{s}\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}K^{-}\pi^{+}\right)}{{\cal
B}\left(B^{0}\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}\pi^{+}\pi^{-}\right)}=1.18\pm
0.05\,\text{(stat.)}\pm 0.12\,\text{(syst.)}\,.$

In addition, the relative branching fraction of the decay
$B^{0}\rightarrow\kern 1.99997pt\overline{\kern-1.99997ptD}{}^{0}K^{+}\pi^{-}$
is measured to be

$\frac{{\cal B}\left(B^{0}\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}K^{+}\pi^{-}\right)}{{\cal
B}\left(B^{0}\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}\pi^{+}\pi^{-}\right)}=0.106\pm
0.007\,\text{(stat.)}\pm 0.008\,\text{(syst.)}\,.$

Submitted to Phys. Rev. D.

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

LHCb collaboration

R. Aaij40, C. Abellan Beteta35,n, B. Adeva36, M. Adinolfi45, C. Adrover6, A.
Affolder51, Z. Ajaltouni5, J. Albrecht9, F. Alessio37, M. Alexander50, S.
Ali40, G. Alkhazov29, P. Alvarez Cartelle36, A.A. Alves Jr24,37, S. Amato2, S.
Amerio21, Y. Amhis7, L. Anderlini17,f, J. Anderson39, R. Andreassen56, R.B.
Appleby53, O. Aquines Gutierrez10, F. Archilli18, A. Artamonov 34, M.
Artuso58, E. Aslanides6, G. Auriemma24,m, S. Bachmann11, J.J. Back47, C.
Baesso59, V. Balagura30, W. Baldini16, R.J. Barlow53, C. Barschel37, S.
Barsuk7, W. Barter46, Th. Bauer40, A. Bay38, J. Beddow50, F. Bedeschi22, I.
Bediaga1, S. Belogurov30, K. Belous34, I. Belyaev30, E. Ben-Haim8, G.
Bencivenni18, S. Benson49, J. Benton45, A. Berezhnoy31, R. Bernet39, M.-O.
Bettler46, M. van Beuzekom40, A. Bien11, S. Bifani44, T. Bird53, A.
Bizzeti17,h, P.M. Bjørnstad53, T. Blake37, F. Blanc38, J. Blouw11, S. Blusk58,
V. Bocci24, A. Bondar33, N. Bondar29, W. Bonivento15, S. Borghi53, A.
Borgia58, T.J.V. Bowcock51, E. Bowen39, C. Bozzi16, T. Brambach9, J. van den
Brand41, J. Bressieux38, D. Brett53, M. Britsch10, T. Britton58, N.H. Brook45,
H. Brown51, I. Burducea28, A. Bursche39, G. Busetto21,q, J. Buytaert37, S.
Cadeddu15, O. Callot7, M. Calvi20,j, M. Calvo Gomez35,n, A. Camboni35, P.
Campana18,37, D. Campora Perez37, A. Carbone14,c, G. Carboni23,k, R.
Cardinale19,i, A. Cardini15, H. Carranza-Mejia49, L. Carson52, K. Carvalho
Akiba2, G. Casse51, L. Castillo Garcia37, M. Cattaneo37, Ch. Cauet9, M.
Charles54, Ph. Charpentier37, P. Chen3,38, N. Chiapolini39, M. Chrzaszcz 25,
K. Ciba37, X. Cid Vidal37, G. Ciezarek52, P.E.L. Clarke49, M. Clemencic37,
H.V. Cliff46, J. Closier37, C. Coca28, V. Coco40, J. Cogan6, E. Cogneras5, P.
Collins37, A. Comerma-Montells35, A. Contu15,37, A. Cook45, M. Coombes45, S.
Coquereau8, G. Corti37, B. Couturier37, G.A. Cowan49, D.C. Craik47, S.
Cunliffe52, R. Currie49, C. D’Ambrosio37, P. David8, P.N.Y. David40, A.
Davis56, I. De Bonis4, K. De Bruyn40, S. De Capua53, M. De Cian39, J.M. De
Miranda1, L. De Paula2, W. De Silva56, P. De Simone18, D. Decamp4, M.
Deckenhoff9, L. Del Buono8, N. Déléage4, D. Derkach14, O. Deschamps5, F.
Dettori41, A. Di Canto11, F. Di Ruscio23,k, H. Dijkstra37, M. Dogaru28, S.
Donleavy51, F. Dordei11, A. Dosil Suárez36, D. Dossett47, A. Dovbnya42, F.
Dupertuis38, R. Dzhelyadin34, A. Dziurda25, A. Dzyuba29, S. Easo48,37, U.
Egede52, V. Egorychev30, S. Eidelman33, D. van Eijk40, S. Eisenhardt49, U.
Eitschberger9, R. Ekelhof9, L. Eklund50,37, I. El Rifai5, Ch. Elsasser39, D.
Elsby44, A. Falabella14,e, C. Färber11, G. Fardell49, C. Farinelli40, S.
Farry51, V. Fave38, D. Ferguson49, V. Fernandez Albor36, F. Ferreira
Rodrigues1, M. Ferro-Luzzi37, S. Filippov32, M. Fiore16, C. Fitzpatrick37, M.
Fontana10, F. Fontanelli19,i, R. Forty37, O. Francisco2, M. Frank37, C.
Frei37, M. Frosini17,f, S. Furcas20, E. Furfaro23,k, A. Gallas Torreira36, D.
Galli14,c, M. Gandelman2, P. Gandini58, Y. Gao3, J. Garofoli58, P. Garosi53,
J. Garra Tico46, L. Garrido35, C. Gaspar37, R. Gauld54, E. Gersabeck11, M.
Gersabeck53, T. Gershon47,37, Ph. Ghez4, V. Gibson46, V.V. Gligorov37, C.
Göbel59, D. Golubkov30, A. Golutvin52,30,37, A. Gomes2, H. Gordon54, M.
Grabalosa Gándara5, R. Graciani Diaz35, L.A. Granado Cardoso37, E. Graugés35,
G. Graziani17, A. Grecu28, E. Greening54, S. Gregson46, P. Griffith44, O.
Grünberg60, B. Gui58, E. Gushchin32, Yu. Guz34,37, T. Gys37, C.
Hadjivasiliou58, G. Haefeli38, C. Haen37, S.C. Haines46, S. Hall52, T.
Hampson45, S. Hansmann-Menzemer11, N. Harnew54, S.T. Harnew45, J. Harrison53,
T. Hartmann60, J. He37, V. Heijne40, K. Hennessy51, P. Henrard5, J.A. Hernando
Morata36, E. van Herwijnen37, E. Hicks51, D. Hill54, M. Hoballah5, C.
Hombach53, P. Hopchev4, W. Hulsbergen40, P. Hunt54, T. Huse51, N. Hussain54,
D. Hutchcroft51, D. Hynds50, V. Iakovenko43, M. Idzik26, P. Ilten12, R.
Jacobsson37, A. Jaeger11, E. Jans40, P. Jaton38, A. Jawahery57, F. Jing3, M.
John54, D. Johnson54, C.R. Jones46, C. Joram37, B. Jost37, M. Kaballo9, S.
Kandybei42, M. Karacson37, T.M. Karbach37, I.R. Kenyon44, U. Kerzel37, T.
Ketel41, A. Keune38, B. Khanji20, O. Kochebina7, I. Komarov38, R.F. Koopman41,
P. Koppenburg40, M. Korolev31, A. Kozlinskiy40, L. Kravchuk32, K. Kreplin11,
M. Kreps47, G. Krocker11, P. Krokovny33, F. Kruse9, M. Kucharczyk20,25,j, V.
Kudryavtsev33, T. Kvaratskheliya30,37, V.N. La Thi38, D. Lacarrere37, G.
Lafferty53, A. Lai15, D. Lambert49, R.W. Lambert41, E. Lanciotti37, G.
Lanfranchi18, C. Langenbruch37, T. Latham47, C. Lazzeroni44, R. Le Gac6, J.
van Leerdam40, J.-P. Lees4, R. Lefèvre5, A. Leflat31, J. Lefrançois7, S.
Leo22, O. Leroy6, T. Lesiak25, B. Leverington11, Y. Li3, L. Li Gioi5, M.
Liles51, R. Lindner37, C. Linn11, B. Liu3, G. Liu37, S. Lohn37, I.
Longstaff50, J.H. Lopes2, E. Lopez Asamar35, N. Lopez-March38, H. Lu3, D.
Lucchesi21,q, J. Luisier38, H. Luo49, F. Machefert7, I.V. Machikhiliyan4,30,
F. Maciuc28, O. Maev29,37, S. Malde54, G. Manca15,d, G. Mancinelli6, U.
Marconi14, R. Märki38, J. Marks11, G. Martellotti24, A. Martens8, L. Martin54,
A. Martín Sánchez7, M. Martinelli40, D. Martinez Santos41, D. Martins Tostes2,
A. Massafferri1, R. Matev37, Z. Mathe37, C. Matteuzzi20, E. Maurice6, A.
Mazurov16,32,37,e, J. McCarthy44, A. McNab53, R. McNulty12, B. Meadows56,54,
F. Meier9, M. Meissner11, M. Merk40, D.A. Milanes8, M.-N. Minard4, J. Molina
Rodriguez59, S. Monteil5, D. Moran53, P. Morawski25, M.J. Morello22,s, R.
Mountain58, I. Mous40, F. Muheim49, K. Müller39, R. Muresan28, B. Muryn26, B.
Muster38, P. Naik45, T. Nakada38, R. Nandakumar48, I. Nasteva1, M. Needham49,
N. Neufeld37, A.D. Nguyen38, T.D. Nguyen38, C. Nguyen-Mau38,p, M. Nicol7, V.
Niess5, R. Niet9, N. Nikitin31, T. Nikodem11, A. Nomerotski54, A. Novoselov34,
A. Oblakowska-Mucha26, V. Obraztsov34, S. Oggero40, S. Ogilvy50, O.
Okhrimenko43, R. Oldeman15,d, M. Orlandea28, J.M. Otalora Goicochea2, P.
Owen52, A. Oyanguren 35,o, B.K. Pal58, A. Palano13,b, M. Palutan18, J.
Panman37, A. Papanestis48, M. Pappagallo50, C. Parkes53, C.J. Parkinson52, G.
Passaleva17, G.D. Patel51, M. Patel52, G.N. Patrick48, C. Patrignani19,i, C.
Pavel-Nicorescu28, A. Pazos Alvarez36, A. Pellegrino40, G. Penso24,l, M. Pepe
Altarelli37, S. Perazzini14,c, D.L. Perego20,j, E. Perez Trigo36, A. Pérez-
Calero Yzquierdo35, P. Perret5, M. Perrin-Terrin6, G. Pessina20, K.
Petridis52, A. Petrolini19,i, A. Phan58, E. Picatoste Olloqui35, B. Pietrzyk4,
T. Pilař47, D. Pinci24, S. Playfer49, M. Plo Casasus36, F. Polci8, G. Polok25,
A. Poluektov47,33, E. Polycarpo2, A. Popov34, D. Popov10, B. Popovici28, C.
Potterat35, A. Powell54, J. Prisciandaro38, V. Pugatch43, A. Puig Navarro38,
G. Punzi22,r, W. Qian4, J.H. Rademacker45, B. Rakotomiaramanana38, M.S.
Rangel2, I. Raniuk42, N. Rauschmayr37, G. Raven41, S. Redford54, M.M. Reid47,
A.C. dos Reis1, S. Ricciardi48, A. Richards52, K. Rinnert51, V. Rives
Molina35, D.A. Roa Romero5, P. Robbe7, E. Rodrigues53, P. Rodriguez Perez36,
S. Roiser37, V. Romanovsky34, A. Romero Vidal36, J. Rouvinet38, T. Ruf37, F.
Ruffini22, H. Ruiz35, P. Ruiz Valls35,o, G. Sabatino24,k, J.J. Saborido
Silva36, N. Sagidova29, P. Sail50, B. Saitta15,d, V. Salustino Guimaraes2, C.
Salzmann39, B. Sanmartin Sedes36, M. Sannino19,i, R. Santacesaria24, C.
Santamarina Rios36, E. Santovetti23,k, M. Sapunov6, A. Sarti18,l, C.
Satriano24,m, A. Satta23, M. Savrie16,e, D. Savrina30,31, P. Schaack52, M.
Schiller41, H. Schindler37, M. Schlupp9, M. Schmelling10, B. Schmidt37, O.
Schneider38, A. Schopper37, M.-H. Schune7, R. Schwemmer37, B. Sciascia18, A.
Sciubba24, M. Seco36, A. Semennikov30, K. Senderowska26, I. Sepp52, N.
Serra39, J. Serrano6, P. Seyfert11, M. Shapkin34, I. Shapoval16,42, P.
Shatalov30, Y. Shcheglov29, T. Shears51,37, L. Shekhtman33, O. Shevchenko42,
V. Shevchenko30, A. Shires52, R. Silva Coutinho47, T. Skwarnicki58, N.A.
Smith51, E. Smith54,48, M. Smith53, M.D. Sokoloff56, F.J.P. Soler50, F.
Soomro18, D. Souza45, B. Souza De Paula2, B. Spaan9, A. Sparkes49, P.
Spradlin50, F. Stagni37, S. Stahl11, O. Steinkamp39, S. Stoica28, S. Stone58,
B. Storaci39, M. Straticiuc28, U. Straumann39, V.K. Subbiah37, L. Sun56, S.
Swientek9, V. Syropoulos41, M. Szczekowski27, P. Szczypka38,37, T. Szumlak26,
S. T’Jampens4, M. Teklishyn7, E. Teodorescu28, F. Teubert37, C. Thomas54, E.
Thomas37, J. van Tilburg11, V. Tisserand4, M. Tobin38, S. Tolk41, D.
Tonelli37, S. Topp-Joergensen54, N. Torr54, E. Tournefier4,52, S. Tourneur38,
M.T. Tran38, M. Tresch39, A. Tsaregorodtsev6, P. Tsopelas40, N. Tuning40, M.
Ubeda Garcia37, A. Ukleja27, D. Urner53, U. Uwer11, V. Vagnoni14, G.
Valenti14, R. Vazquez Gomez35, P. Vazquez Regueiro36, S. Vecchi16, J.J.
Velthuis45, M. Veltri17,g, G. Veneziano38, M. Vesterinen37, B. Viaud7, D.
Vieira2, X. Vilasis-Cardona35,n, A. Vollhardt39, D. Volyanskyy10, D. Voong45,
A. Vorobyev29, V. Vorobyev33, C. Voß60, H. Voss10, R. Waldi60, R. Wallace12,
S. Wandernoth11, J. Wang58, D.R. Ward46, N.K. Watson44, A.D. Webber53, D.
Websdale52, M. Whitehead47, J. Wicht37, J. Wiechczynski25, D. Wiedner11, L.
Wiggers40, G. Wilkinson54, M.P. Williams47,48, M. Williams55, F.F. Wilson48,
J. Wishahi9, M. Witek25, S.A. Wotton46, S. Wright46, S. Wu3, K. Wyllie37, Y.
Xie49,37, F. Xing54, Z. Xing58, Z. Yang3, R. Young49, X. Yuan3, O.
Yushchenko34, M. Zangoli14, M. Zavertyaev10,a, F. Zhang3, L. Zhang58, W.C.
Zhang12, Y. Zhang3, A. Zhelezov11, A. Zhokhov30, L. Zhong3, A. Zvyagin37.

1Centro Brasileiro de Pesquisas Físicas (CBPF), Rio de Janeiro, Brazil
2Universidade Federal do Rio de Janeiro (UFRJ), Rio de Janeiro, Brazil
3Center for High Energy Physics, Tsinghua University, Beijing, China
4LAPP, Université de Savoie, CNRS/IN2P3, Annecy-Le-Vieux, France
5Clermont Université, Université Blaise Pascal, CNRS/IN2P3, LPC, Clermont-
Ferrand, France
6CPPM, Aix-Marseille Université, CNRS/IN2P3, Marseille, France
7LAL, Université Paris-Sud, CNRS/IN2P3, Orsay, France
8LPNHE, Université Pierre et Marie Curie, Université Paris Diderot,
CNRS/IN2P3, Paris, France
9Fakultät Physik, Technische Universität Dortmund, Dortmund, Germany
10Max-Planck-Institut für Kernphysik (MPIK), Heidelberg, Germany
11Physikalisches Institut, Ruprecht-Karls-Universität Heidelberg, Heidelberg,
Germany
12School of Physics, University College Dublin, Dublin, Ireland
13Sezione INFN di Bari, Bari, Italy
14Sezione INFN di Bologna, Bologna, Italy
15Sezione INFN di Cagliari, Cagliari, Italy
16Sezione INFN di Ferrara, Ferrara, Italy
17Sezione INFN di Firenze, Firenze, Italy
18Laboratori Nazionali dell’INFN di Frascati, Frascati, Italy
19Sezione INFN di Genova, Genova, Italy
20Sezione INFN di Milano Bicocca, Milano, Italy
21Sezione INFN di Padova, Padova, Italy
22Sezione INFN di Pisa, Pisa, Italy
23Sezione INFN di Roma Tor Vergata, Roma, Italy
24Sezione INFN di Roma La Sapienza, Roma, Italy
25Henryk Niewodniczanski Institute of Nuclear Physics Polish Academy of
Sciences, Kraków, Poland
26AGH - University of Science and Technology, Faculty of Physics and Applied
Computer Science, Kraków, Poland
27National Center for Nuclear Research (NCBJ), Warsaw, Poland
28Horia Hulubei National Institute of Physics and Nuclear Engineering,
Bucharest-Magurele, Romania
29Petersburg Nuclear Physics Institute (PNPI), Gatchina, Russia
30Institute of Theoretical and Experimental Physics (ITEP), Moscow, Russia
31Institute of Nuclear Physics, Moscow State University (SINP MSU), Moscow,
Russia
32Institute for Nuclear Research of the Russian Academy of Sciences (INR RAN),
Moscow, Russia
33Budker Institute of Nuclear Physics (SB RAS) and Novosibirsk State
University, Novosibirsk, Russia
34Institute for High Energy Physics (IHEP), Protvino, Russia
35Universitat de Barcelona, Barcelona, Spain
36Universidad de Santiago de Compostela, Santiago de Compostela, Spain
37European Organization for Nuclear Research (CERN), Geneva, Switzerland
38Ecole Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland
39Physik-Institut, Universität Zürich, Zürich, Switzerland
40Nikhef National Institute for Subatomic Physics, Amsterdam, The Netherlands
41Nikhef National Institute for Subatomic Physics and VU University Amsterdam,
Amsterdam, The Netherlands
42NSC Kharkiv Institute of Physics and Technology (NSC KIPT), Kharkiv, Ukraine
43Institute for Nuclear Research of the National Academy of Sciences (KINR),
Kyiv, Ukraine
44University of Birmingham, Birmingham, United Kingdom
45H.H. Wills Physics Laboratory, University of Bristol, Bristol, United
Kingdom
46Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom
47Department of Physics, University of Warwick, Coventry, United Kingdom
48STFC Rutherford Appleton Laboratory, Didcot, United Kingdom
49School of Physics and Astronomy, University of Edinburgh, Edinburgh, United
Kingdom
50School of Physics and Astronomy, University of Glasgow, Glasgow, United
Kingdom
51Oliver Lodge Laboratory, University of Liverpool, Liverpool, United Kingdom
52Imperial College London, London, United Kingdom
53School of Physics and Astronomy, University of Manchester, Manchester,
United Kingdom
54Department of Physics, University of Oxford, Oxford, United Kingdom
55Massachusetts Institute of Technology, Cambridge, MA, United States
56University of Cincinnati, Cincinnati, OH, United States
57University of Maryland, College Park, MD, United States
58Syracuse University, Syracuse, NY, United States
59Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), Rio de
Janeiro, Brazil, associated to 2
60Institut für Physik, Universität Rostock, Rostock, Germany, associated to 11

aP.N. Lebedev Physical Institute, Russian Academy of Science (LPI RAS),
Moscow, Russia
bUniversità di Bari, Bari, Italy
cUniversità di Bologna, Bologna, Italy
dUniversità di Cagliari, Cagliari, Italy
eUniversità di Ferrara, Ferrara, Italy
fUniversità di Firenze, Firenze, Italy
gUniversità di Urbino, Urbino, Italy
hUniversità di Modena e Reggio Emilia, Modena, Italy
iUniversità di Genova, Genova, Italy
jUniversità di Milano Bicocca, Milano, Italy
kUniversità di Roma Tor Vergata, Roma, Italy
lUniversità di Roma La Sapienza, Roma, Italy
mUniversità della Basilicata, Potenza, Italy
nLIFAELS, La Salle, Universitat Ramon Llull, Barcelona, Spain
oIFIC, Universitat de Valencia-CSIC, Valencia, Spain
pHanoi University of Science, Hanoi, Viet Nam
qUniversità di Padova, Padova, Italy
rUniversità di Pisa, Pisa, Italy
sScuola Normale Superiore, Pisa, Italy

##  1 Introduction

The precise measurement of the angle $\gamma$ of the CKM Unitarity Triangle
[1, 2] is one of the primary objectives in contemporary flavour physics.
Measurements from the experiments BaBar, Belle and LHCb are based mainly on
studies of $B^{+}\rightarrow DK^{+}$ decays, where the notation $D$ implies
that the neutral $D$ meson is an admixture of $D^{0}$ and $\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}$ states. Each experiment currently
gives constraints on $\gamma$ with a precision of $\sim 15^{\circ}$ [3, 4, 5].
Significant reduction of this uncertainty is well motivated and the use of
additional channels to further improve the precision is of great interest.

The decay $B^{0}\rightarrow DK^{+}\pi^{-}$, including the resonant
contribution from $B^{0}\rightarrow DK^{*0}$, is one of the modes with the
potential to make significant impact on the overall determination of $\gamma$
[6]. A first measurement of $C\\!P$ observables in $B^{0}\rightarrow DK^{*0}$
decays has been reported by LHCb [7]. This decay is particularly sensitive to
$\gamma$ owing to the interference of $b\rightarrow c\bar{u}s$ and
$b\rightarrow u\bar{c}s$ amplitudes, which for this decay are of similar
magnitude. It has been noted that an amplitude analysis of $B^{0}\rightarrow
DK^{+}\pi^{-}$ decays can further improve the sensitivity and also resolve the
ambiguities in the result [8, 9].

The decays $B^{0}\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}K^{+}\pi^{-}$ and
$B^{0}_{s}\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}K^{-}\pi^{+}$ can be mediated by the
decay diagrams shown in Fig. 1. Both $B^{0}$ and $B^{0}_{s}$ decays are
flavour-specific, with the charge of the kaon identifying the flavour of the
decaying $B$ meson, though the charges are opposite in the two cases. In
addition to these colour-allowed tree-level diagrams, colour-suppressed tree-
level diagrams contribute to $B^{0}_{(s)}\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}K\pi$ decays ($K\pi$ denotes the sum
over both charge combinations). Both colour-allowed and colour-suppressed
diagrams contribute to the CKM-suppressed $B^{0}_{(s)}\rightarrow D^{0}K\pi$
modes.

Figure 1:  Decay diagrams for (a) favoured $B^{0}\rightarrow\kern
1.79997pt\overline{\kern-1.79997ptD}{}^{0}K^{+}\pi^{-}$ decays and (b)
favoured $B^{0}_{s}\rightarrow\kern
1.79997pt\overline{\kern-1.79997ptD}{}^{0}K^{-}\pi^{+}$ decays.

A first study of the decay $B^{0}\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}K^{+}\pi^{-}$ has been performed by
BaBar [10], giving a branching fraction measurement ${\cal
B}\left(B^{0}\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}K^{+}\pi^{-}\right)=(88\pm 15\pm
9)\times 10^{-6}$, where the contribution from the $B^{0}\rightarrow
D^{*-}K^{+}$ decay is excluded. There is no previous branching fraction
measurement for the inclusive three-body process $B^{0}_{s}\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}K^{-}\pi^{+}$, although that of the
resonant contribution $\kern 1.99997pt\overline{\kern-1.99997ptD}{}^{0}\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$ has been measured by LHCb [11].
Since the $B^{0}_{s}\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}K^{-}\pi^{+}$ and the related
$B^{0}_{s}\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{*0}K^{-}\pi^{+}$ decays form
potentially serious backgrounds to the $B^{0}$ $\rightarrow$ $D$ $K^{+}$
$\pi^{-}$ channel, measurements of their properties will be necessary to
reduce systematic uncertainties in the determination of $\gamma$.

In this paper the results of a study of neutral $B$ meson decays to $\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}K\pi$, including inspections of
their Dalitz plot distributions, are presented. The $\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}K^{+}\pi^{-}$ and $\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}K^{-}\pi^{+}$ final states are
combined, and the inclusion of charge conjugate processes is implied
throughout the paper. In order to reduce systematic uncertainties in the
measurements, the topologically similar decay $\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}\pi^{+}\pi^{-}$, which has been
studied in detail previously [12, 13], is used as a normalisation channel. In
this paper, $D\pi\pi$ denotes the $\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}\pi^{+}\pi^{-}$ final state and
$DK\pi$ denotes the sum over the $\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}K^{+}\pi^{-}$ and $\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}K^{-}\pi^{+}$ final states. The
neutral $D$ meson is reconstructed using the $\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}\rightarrow K^{+}\pi^{-}$ final
state; therefore the signal yields measured include small contributions from
$D^{0}\rightarrow K^{+}\pi^{-}$ decays, but such contributions are expected to
be small and are neglected hereafter. The analysis uses a data sample,
corresponding to an integrated luminosity of $1.0\mbox{\,fb}^{-1}$ of $pp$
collisions at a centre-of-mass energy of $7\mathrm{\,Te\kern-1.00006ptV}$,
collected with the LHCb detector during 2011.

##  2 Detector, trigger and selection

The LHCb detector [14] 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
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
(IP) resolution of 20$\,\upmu\rm m$ for tracks with high transverse momentum
($p_{\rm T}$). Charged hadrons are identified using two ring-imaging Cherenkov
(RICH) detectors [15]. 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 LHCb trigger [16] consists of a hardware stage, based on information from
the calorimeter and muon systems, followed by a software stage that applies a
full event reconstruction. In this analysis, signal candidates are accepted if
one of the final state particles created a cluster in the hadronic calorimeter
with sufficient transverse energy to fire the hardware trigger. Events that
are triggered at the hardware level by another particle in the event are also
retained.

The software trigger requires a two-, three- or four-track secondary vertex
with a high sum of the transverse momentum, $p_{\rm T}$, 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 impact parameter $\chi^{2}$,
$\chi^{2}_{\rm IP}$, with respect to the primary interaction greater than 16.
The $\chi^{2}_{\rm IP}$ is the difference between the $\chi^{2}$ of the PV
reconstruction with and without the considered track. A multivariate algorithm
[17] is used for the identification of secondary vertices consistent with the
decay of a $b$ hadron.

Candidates that satisfy the software trigger selection and are consistent with
the decay chain $B^{0}_{(s)}\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}K^{\pm}\pi^{\mp}$, $\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}\rightarrow K^{+}\pi^{-}$ are
selected, with requirements similar to those in the LHCb study of the decay
$B^{0}_{(s)}\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}K^{+}K^{-}$ [18]. The $\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}$ candidate invariant mass is
required to satisfy
$1844<m_{K\pi}<1884{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$. Tracks are
required to be consistent with either the kaon or pion hypothesis, as
appropriate, based on particle identification (PID) information primarily from
the RICH detectors [15]. All other selection criteria were tuned on the $\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}\pi^{+}\pi^{-}$ channel. The large
yield available for the $B^{0}\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}\pi^{+}\pi^{-}$ normalisation sample
allows the selection to be based on data, though the efficiencies are
determined using simulated events. In the simulation, $pp$ collisions are
generated using Pythia 6.4 [19] with a specific LHCb configuration [20].
Decays of hadronic particles are described by EvtGen [21] in which final state
radiation is generated using Photos [22]. The interaction of the generated
particles with the detector and its response are implemented using the Geant4
toolkit [23, *Agostinelli:2002hh] as described in Ref. [25].

Loose selection requirements are applied to obtain a visible signal peak in
the $\kern 1.99997pt\overline{\kern-1.99997ptD}{}^{0}\pi^{+}\pi^{-}$
normalisation channel. The selection includes criteria on the quality of the
tracks forming the signal candidate, their $p$, $p_{\rm T}$ and $\chi^{2}_{\rm
IP}$. Requirements are also placed on the corresponding variables for
candidate composite particles ($\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}$, $B^{0}_{(s)}$) together with
restrictions on the consistency of the decay fit ($\chi^{2}_{\rm vertex}$),
the flight distance significance ($\chi^{2}_{\rm flight}$), and the cosine of
the angle between the momentum vector and the line joining the PV under
consideration to the $B^{0}_{(s)}$ vertex ($\cos\theta_{\rm dir}$) [11].

A boosted decision tree (BDT) [26] that identifies $\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}\rightarrow K^{+}\pi^{-}$ candidates
is used to suppress backgrounds from $b$-hadron decays to final states that do
not contain charmed particles and backgrounds where the $\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}$ does not decay to the
$K^{+}\pi^{-}$ final state. This “$D^{0}$ BDT” [27, 28] is trained using a
large high-purity sample obtained from $B^{+}\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}\pi^{+}$ decays. The BDT takes
advantage of the kinematic similarity of all $b$-hadron decays and avoids
using any topological information from the $B^{0}_{(s)}$ decay. Properties of
the $\kern 1.99997pt\overline{\kern-1.99997ptD}{}^{0}$ candidate and its
daughter tracks, containing kinematic, track quality, vertex and PID
information, are used to train the BDT.

Further discrimination between signal and background categories is achieved by
calculating weights, using the sPlot technique [29], for the remaining $\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}\pi^{+}\pi^{-}$ candidates. The
weights are based on a simplified fit to the $B$ candidate invariant mass
distribution from the $D\pi\pi$ data sample. The weights are used to train a
neural network [30] to maximise the separation between the categories. A total
of 10 variables are used in the network. They include the $p_{\rm T}$,
$\chi^{2}_{\rm IP}$, $\chi^{2}_{\rm vertex}$, $\chi^{2}_{\rm flight}$ and
$\cos\theta_{\rm dir}$ of the $B^{0}_{(s)}$ candidate, the output of the
$D^{0}$ BDT and the $\chi^{2}_{\rm IP}$ of the two pion tracks that originate
from the $B^{0}_{(s)}$ vertex. The $p_{\rm T}$ asymmetry and track
multiplicity in a cone with half-angle of 1.5 units in the plane of
pseudorapidity and azimuthal angle (measured in radians) [31] around the
$B^{0}_{(s)}$ candidate flight direction are also used. The input quantities
to the neural network only depend weakly on the kinematics of the
$B^{0}_{(s)}$ decay. A requirement on the network output is imposed that
reduces the combinatorial background by an order of magnitude while retaining
about 70 % of the signal.

To improve the $B^{0}_{(s)}$ candidate invariant mass resolution, the four-
momenta of the tracks from the $\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}$ candidate are adjusted [32] so
that their combined invariant mass matches the world average value [33]. An
additional $B^{0}_{(s)}$ mass constraint is applied in the calculation of the
Dalitz plot coordinates, $m^{2}(DK)$ and $m^{2}(D\pi)$, which are used in the
determination of event-by-event efficiencies. The coordinates are calculated
twice: once each with a $B^{0}$ and a $B^{0}_{s}$ mass constraint. A small
fraction ($\sim 1\,\%$ within the fitted mass range) of candidates with
invariant masses far from the $B^{0}_{(s)}$ peak fail one or both of these
mass-constrained fits, and are removed from the analysis.

To remove the large background from $B^{0}\rightarrow D^{*-}\pi^{+}$ decays,
candidates in both samples are rejected if the mass difference
$m_{D\pi}$–$m_{D}$ (for either pion charge in the combinations $\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}\pi^{+}\pi^{-}$ and $\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}K\pi$) lies within $\pm
2.5{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ of the nominal $D^{*-}$–$\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}$ mass difference [33]. Candidates
in the $DK\pi$ sample are also rejected if the mass difference
$m_{DK}$–$m_{D}$ calculated under the pion mass hypothesis satisfies the same
criterion. A potential background contribution from $B^{0}_{s}\rightarrow
D^{\mp}K^{\pm}$ decays is removed by requiring that the pion from the $\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}$ candidate together with the kaon
and the pion do not form an invariant mass in the range
$1850$–$1885{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$. Further $DK\pi$
candidates are rejected by requiring that the kaon from the $\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}$ candidate together with the kaon
and the pion do not form an invariant mass in the range
$1955$–$1975{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$, which removes potential
background from $B^{0}_{s}\rightarrow D^{\mp}_{s}\pi^{\pm}$ decays. A muon
veto is applied to all four final state tracks to remove potential background
from $B^{0}_{(s)}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{*0}$ decays and $\kern 1.99997pt\overline{\kern-1.99997ptD}{}^{0}$
candidates are required to travel at least $1\rm\,mm$ from the $B^{0}_{(s)}$
decay vertex to remove charmless backgrounds that survive the $D^{0}$ BDT
requirement.

Candidates are retained for further analysis if they have an invariant mass in
the range $5150$–$5600{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ for $D\pi\pi$
or $5200$–$5600{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ for $DK\pi$. After
all selection requirements are applied, fewer than 1 % of events with at least
one candidate also contain a second candidate. Such multiple candidates are
retained and treated in the same manner as other candidates; the associated
systematic uncertainty is negligible.

##  3 Determination of signal yields

The signal yields are obtained from unbinned maximum likelihood fits to the
invariant mass distributions. In addition to signal contributions and
combinatorial background, candidates may be formed from misidentified or
partially reconstructed $b$-hadron decays. Contributions from partially
reconstructed decays are reduced by the lower bounds on the invariant mass
regions used in the fits. Sources of misidentified backgrounds are
investigated using simulation. Most potential sources are found to have broad
invariant mass distributions, and are absorbed in the combinatorial background
shapes used in the fits described below. Backgrounds from $\kern
1.00006pt\overline{\kern-1.00006pt\mathchar
28931\relax}^{0}_{b}\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}\overline{}p\pi^{+}$ [34] and
$B^{0}\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}\pi^{+}\pi^{-}$ decays may, however,
give contributions with distinctive shapes in the mass distributions of
$D\pi\pi$ and $DK\pi$ candidates, respectively, and are therefore explicitly
modelled in the fits.

The $D\pi\pi$ fit includes a double Gaussian shape to describe the signal,
where the two Gaussian functions share a common mean, together with an
exponential component for partially reconstructed background, and a
probability density function (PDF) for $\kern
1.00006pt\overline{\kern-1.00006pt\mathchar
28931\relax}^{0}_{b}\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}\overline{}p\pi^{+}$ decays. This
PDF is modelled using a smoothed non-parametric function obtained from
simulated data, reweighted so that the $\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}\pi^{+}$ invariant mass distribution
matches that observed in data. The shape of the combinatorial background is
essentially linear, but is multiplied by a function that accounts for the fact
that candidates with high invariant masses are more likely to fail the
$B^{0}_{(s)}$ mass constrained fit. There are ten free parameters in the
$D\pi\pi$ fit: the double Gaussian peak position, the widths of the two
Gaussian shapes and the relative normalisation of the two Gaussian functions,
the linear slope of the combinatorial background, the exponential shape
parameter of the partially reconstructed background, and the yields of the
four categories. The result of the fit to the $D\pi\pi$ candidates is shown in
Fig. 2(a) and yields $8558\pm 134$ $B^{0}\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}\pi^{+}\pi^{-}$ decays.

Figure 2:  Fits to the $B^{0}_{(s)}$ candidate invariant mass distributions
for the (a) $D\pi\pi$ and (b) $DK\pi$ samples. Data points are shown in black,
the full fitted PDFs as solid blue lines and the components as detailed in the
legends.

The $DK\pi$ fit includes a second double Gaussian component to account for the
presence of both $B^{0}$ and $B^{0}_{s}$ decays. The peaking background PDF
for $B^{0}\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}\pi^{+}\pi^{-}$ decays is modelled
using a smoothed non-parametric function derived from simulation, reweighted
in the same way as described for $\kern
1.00006pt\overline{\kern-1.00006pt\mathchar
28931\relax}^{0}_{b}\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}\overline{}p\pi^{+}$ decays above.
The dominant partially reconstructed backgrounds in the $DK\pi$ fit are from
$B^{0}_{s}$ decays and these extend into the $B^{0}$ signal region. Instead of
an exponential component, a background PDF for $B^{0}_{s}\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{*0}K^{-}\pi^{+}$ decays is included,
modelled using a smoothed non-parametric function obtained from simulation.
Studies using simulated data show that this function can account for all
resonant contributions to the $B^{0}_{s}\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{*0}K^{-}\pi^{+}$ final state. The
function describing the combinatorial background has the same form as for the
$D\pi\pi$ fit. The $DK\pi$ fit has eight free parameters; the parameters of
the double Gaussian functions are constrained to be identical for the $B^{0}$
and $B^{0}_{s}$ signals, with an offset in their mean values fixed to the
known $B^{0}$–$B^{0}_{s}$ mass difference [33]. The relative width of the
broader to the narrower Gaussian component and the relative normalisation of
the two Gaussian functions are constrained within their uncertainties to the
values obtained in simulation. The result of the fit is shown in Fig. 2(b) and
yields $815\pm 55$ $B^{0}\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}K^{+}\pi^{-}$ and $2391\pm 81$
$B^{0}_{s}\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}K^{-}\pi^{+}$ decays. All background
yields in both fits are consistent with their expectations within
uncertainties, based on measured or predicted production rates and branching
fractions and background rejection factors determined from simulations.

##  4 Calculation of branching fraction ratios

The ratios of branching fractions are obtained after applying event-by-event
efficiencies as a function of the Dalitz plot position. The branching fraction
for the $B^{0}\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}K^{+}\pi^{-}$ decay is determined as

$R_{B^{0}}\equiv\frac{{\cal B}\left(B^{0}\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}K^{+}\pi^{-}\right)}{{\cal
B}\left(B^{0}\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}\pi^{+}\pi^{-}\right)}=\frac{N^{\rm
corr}(B^{0}\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}K^{+}\pi^{-})}{N^{\rm
corr}(B^{0}\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}\pi^{+}\pi^{-})}\,,$ (1)

and the branching fraction of the $B^{0}_{s}\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}K^{-}\pi^{+}$ mode is determined as

$R_{B^{0}_{s}}\equiv\frac{{\cal B}\left(B^{0}_{s}\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}K^{-}\pi^{+}\right)}{{\cal
B}\left(B^{0}\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}\pi^{+}\pi^{-}\right)}=\left(\frac{f_{s}}{f_{d}}\right)^{-1}\frac{N^{\rm
corr}(B^{0}_{s}\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}K^{-}\pi^{+})}{N^{\rm
corr}(B^{0}\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}\pi^{+}\pi^{-})}\,,$ (2)

where the efficiency corrected yield is $N^{\rm
corr}=\sum_{i}W_{i}/\epsilon^{\rm tot}_{i}$. Here the index $i$ runs over all
candidates in the fit range, $W_{i}$ is the signal weight for candidate $i$,
determined using the procedure described in Ref. [29], from the fits shown in
Fig. 2 and $\epsilon^{\rm tot}_{i}$ is the efficiency for candidate $i$ as a
function of its Dalitz plot position. The ratio of fragmentation fractions is
$f_{s}/f_{d}=0.256\pm 0.020$ [35]. The statistical uncertainty on the
branching fraction ratio incorporates the effects of the shape parameters that
are allowed to vary in the fit and the dilution due to event weighting. Most
potential systematic effects cancel in the ratio.

The PID efficiency is measured using a control sample of
$D^{*-}\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}\pi^{-},\,\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}\rightarrow K^{+}\pi^{-}$ decays to
obtain background-subtracted efficiency tables for kaons and pions as a
function of their $p$ and $p_{\rm T}$ [36, 15]. The kinematic properties of
the particles in signal decays are obtained from simulation in which events
are uniformly distributed across the phase space, allowing the PID efficiency
for each event to be obtained from the tables, while taking into account the
correlation between the $p$ and $p_{\rm T}$ values of the two tracks. The
other contributions to the efficiency (detector acceptance, selection criteria
and trigger effects) are determined from phase space simulation, and validated
using data. All are found to be approximately constant across the Dalitz
plane, apart from some modulations seen near the kinematic boundaries and, for
the $DK\pi$ channels, a variation caused by different PID requirements on the
pion and the kaon. The efficiency for each mode, averaged across the Dalitz
plot, is given in Table 1 together with the contributions from geometrical
acceptance, trigger and selection requirements and particle identification.

Table 1: Summary of the efficiencies for $D\pi\pi$ and $DK\pi$ in phase space simulation. Contributions from geometrical acceptance ($\epsilon^{\rm geom}$), trigger and selection requirements ($\epsilon^{\rm trig\&sel}$) and particle identification ($\epsilon^{\rm PID}$) are shown. The geometrical acceptance is evaluated for $B$ mesons produced within the detector acceptance. Values given are in percent.  | $B^{0}\rightarrow D\pi\pi$ | $B^{0}\rightarrow DK\pi$ | $B^{0}_{s}\rightarrow DK\pi$
---|---|---|---
$\epsilon^{\rm geom}$ | 44.70 | 46.60 | 46.50
$\epsilon^{\rm trig\&sel}$ |  01.32 |  01.25 |  01.25
$\epsilon^{\rm PID}$ | 89.30 | 74.80 | 75.00
$\epsilon^{\rm tot}$ |  00.53 |  00.44 |  00.44

The Dalitz plots obtained from the signal weights are shown in Fig. 3. The
$B^{0}\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}\pi^{+}\pi^{-}$ plot, Fig. 3(a),
shows contributions from the $\rho^{0}(770)$ and $f_{2}(1270)$ resonances
(upper diagonal edge of the Dalitz plot) and from the $D_{2}^{*-}(2460)$ state
(horizontal band), as expected from previous studies of this decay [12, 13].
The $B^{0}\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}K^{+}\pi^{-}$ plot, Fig. 3(b), shows
contributions from the $K^{*0}(892)$ (upper diagonal edge) and from the
$D_{2}^{*-}(2460)$ (vertical band) resonances, also as expected [10]. The
$B^{0}_{s}\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}K^{-}\pi^{+}$ plot, Fig. 3(c), shows
contributions from the $\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}(892)$ (upper diagonal edge) and
from the $D_{s2}^{*-}(2573)$ (horizontal band) states. The former contribution
is as expected [11]. The decay $B^{0}_{s}\rightarrow D_{s2}^{*-}(2573)\pi^{+}$
has not been observed previously but is expected to exist given the
observation of the $B^{0}_{s}\rightarrow D_{s2}^{*-}(2573)\mu^{+}\nu X$ decay
[37].

Figure 3:  Efficiency corrected Dalitz plot distributions for (a)
$B^{0}\rightarrow\kern
1.79997pt\overline{\kern-1.79997ptD}{}^{0}\pi^{+}\pi^{-}$, (b)
$B^{0}\rightarrow\kern 1.79997pt\overline{\kern-1.79997ptD}{}^{0}K^{+}\pi^{-}$
and (c) $B^{0}_{s}\rightarrow\kern
1.79997pt\overline{\kern-1.79997ptD}{}^{0}K^{-}\pi^{+}$ candidates obtained
from the signal weights.

##  5 Systematic uncertainties and cross-checks

Systematic uncertainties are assigned to both branching fraction ratios due to
the following sources (summarised in Table 2). Note that all uncertainties are
relative. The variation of efficiency across the Dalitz plot may not be
correctly modelled in simulation. A two-dimensional polynomial is used to fit
the variation across the Dalitz region of each of the four contributions to
the efficiency (detector acceptance, selection criteria, PID and trigger
effects). These polynomials are used to generate 1000 simulated pseudo-
experiments, varying the fit parameters within their uncertainties. Each set
of simulations is used to calculate the efficiency corrected yield. The
standard deviation from a Gaussian fit to these yields is used to provide a
systematic uncertainty for each decay mode. This leads to a systematic
uncertainty of 3.4 % (3.1 %) for $R_{B^{0}}$ ($R_{B^{0}_{s}}$). The $DK\pi$
fit model is varied by scaling the signal PDF width ratio to account for the
different masses of the $B^{0}$ and $B^{0}_{s}$ mesons, replacing the PDFs of
the background components with unsmoothed versions, adding components for
potential background from $B^{0}_{s}\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{*0}\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$ and $\kern
1.00006pt\overline{\kern-1.00006pt\mathchar
28931\relax}^{0}_{b}\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}\overline{}p\pi^{+}$ decays, and
replacing the double Gaussian signal components with double Crystal Ball [38]
functions. The $D\pi\pi$ fit model is varied by replacing the PDF of the
$\kern 1.00006pt\overline{\kern-1.00006pt\mathchar
28931\relax}^{0}_{b}\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}\overline{}p\pi^{+}$ component with
an unsmoothed version, varying the slope of the combinatorial background and
replacing the exponential partially reconstructed background component with a
PDF for $B^{0}\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{*0}\pi^{+}\pi^{-}$ decays. Combined in
quadrature, these contribute 6.3 % (4.3 %) to $R_{B^{0}}$ ($R_{B^{0}_{s}}$).
Variations in the $D^{*\pm}$, $D^{\pm}$ and $D^{\pm}_{s}$ vetoes contribute to
$R_{B^{0}}$ ($R_{B^{0}_{s}}$), at the level of $<$0.1 %, 2.0 % and 0.2 % (1.0
%, 0.5 % and 0.2 %), respectively. In addition, the possible differences in
the data to simulation ratios of trigger and PID efficiencies between the two
channels (both 1.0 %) and the limited statistics of the simulated data samples
used to calculate efficiencies (2.0 %) affect both $R_{B^{0}}$ and
$R_{B^{0}_{s}}$. The uncertainty on the quantity $f_{s}/f_{d}$ (7.8 %) affects
only $R_{B^{0}_{s}}$. The total systematic uncertainties are obtained as the
quadratic sums of all contributions.

Table 2:  Systematic uncertainties on $R_{B^{0}}$ and $R_{B^{0}_{s}}$. The total is obtained from the sum in quadrature of all contributions. Note that all uncertainties are relative.  | Uncertainty (%)
---|---
Source |  $<$ $B^{0}$ | $B^{0}_{s}$
Modelling of efficiency |  $<$ 3.4 | 3.1
Fit model |  $<$ 6.3 | 4.3
$D^{*\pm}$ veto |  $<$ 0.1 | 1.0
$D^{\pm}$ veto |  $<$ 2.0 | 0.2
$D^{\pm}_{s}$ veto |  $<$ 0.2 | 0.5
Trigger |  $<$ 1.0 | 1.0
Particle identification |  $<$ 1.0 | 1.0
Simulation statistics |  $<$ 2.0 | 2.0
$f_{s}/f_{d}$ |  $<$ – | 7.8
Total |  $<$ 7.8 | 9.8

A number of cross-checks are performed to test the stability of the results.
Based upon the hardware trigger decision, candidates are separated into three
groups: events in which a particle from the signal decay created a cluster
with enough energy in the calorimeter to fire the trigger, events that were
triggered independently of the signal decay and those events that were
triggered by both the signal decay and the rest of the event. The data sample
is divided by dipole magnet polarity. The neural network and PID requirements
are both tightened and loosened. The PID efficiency is evaluated using the
kinematic properties from $\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}\pi^{+}\pi^{-}$ data instead of from
simulation. The requirement for the $B^{0}_{(s)}$ mass constrained fits to
converge is removed. All cross-checks give consistent results.

##  6 Results and conclusions

In summary, the decay $B^{0}_{s}\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}K^{-}\pi^{+}$ has been observed for
the first time, and its branching fraction relative to that of the
$B^{0}\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}\pi^{+}\pi^{-}$ decay is measured to
be

$\frac{{\cal B}\left(B^{0}_{s}\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}K^{-}\pi^{+}\right)}{{\cal
B}\left(B^{0}\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}\pi^{+}\pi^{-}\right)}=1.18\pm
0.05\,\text{(stat.)}\pm 0.12\,\text{(syst.)}\,.$

The current world average value of ${\cal B}\left(B^{0}\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}\pi^{+}\pi^{-}\right)=(8.4\pm 0.4\pm
0.8)\times 10^{-4}$ [12] assumes equal production of $B^{+}B^{-}$ and
$B^{0}\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ at the $\mathchar
28935\relax{(4S)}$ resonance and uses the $D^{0}$ branching fraction ${\cal
B}\left(D^{0}\rightarrow K^{-}\pi^{+}\right)=(3.80\pm 0.07)\,\%$. Using the
current world average values of $\Gamma(\mathchar 28935\relax{(4S)}\rightarrow
B^{+}B^{-})/\Gamma(\mathchar 28935\relax{(4S)}\rightarrow B^{0}\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0})=1.055\pm 0.025$ [33] and ${\cal
B}\left(D^{0}\rightarrow K^{-}\pi^{+}\right)=(3.88\pm 0.05)\,\%$ [33], the
branching fraction of the normalisation channel becomes ${\cal
B}\left(B^{0}\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}\pi^{+}\pi^{-}\right)=(8.5\pm 0.4\pm
0.8)\times 10^{-4}$. This corrected value gives

${\cal B}\left(B^{0}_{s}\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}K^{-}\pi^{+}\right)=(1.00\pm
0.04\,\text{(stat.)}\pm 0.10\,\text{(syst.)}\pm 0.10\,\text{(}{\cal
B}\text{)})\times 10^{-3}\,,$

where the third uncertainty arises from ${\cal B}\left(B^{0}\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}\pi^{+}\pi^{-}\right)$. The
$B^{0}\rightarrow\kern 1.99997pt\overline{\kern-1.99997ptD}{}^{0}K^{+}\pi^{-}$
decay has also been measured, with relative branching fraction

$\frac{{\cal B}\left(B^{0}\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}K^{+}\pi^{-}\right)}{{\cal
B}\left(B^{0}\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}\pi^{+}\pi^{-}\right)}=0.106\pm
0.007\,\text{(stat.)}\pm 0.008\,\text{(syst.)}\,.$

Using the corrected value of ${\cal B}\left(B^{0}\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}\pi^{+}\pi^{-}\right)$ gives

${\cal B}\left(B^{0}\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}K^{+}\pi^{-}\right)=(9.0\pm
0.6\,\text{(stat.)}\pm 0.7\,\text{(syst.)}\pm 0.9\,\text{(}{\cal
B}\text{)})\times 10^{-5}\,,$

which is the most precise measurement of this quantity to date. Future studies
of the Dalitz plot distributions of these decays will provide insight into the
dynamics of hadronic $B$ decays. In addition, the $B^{0}\rightarrow
DK^{+}\pi^{-}$ decay may be used to measure the $C\\!P$ violating phase
$\gamma$.

## Acknowledgements

We express our gratitude to our colleagues in the CERN accelerator departments
for the excellent performance of the LHC. We thank the technical and
administrative staff at the LHCb institutes. We acknowledge support from CERN
and from the national agencies: CAPES, CNPq, FAPERJ and FINEP (Brazil); NSFC
(China); CNRS/IN2P3 and Region Auvergne (France); BMBF, DFG, HGF and MPG
(Germany); SFI (Ireland); INFN (Italy); FOM and NWO (The Netherlands); SCSR
(Poland); ANCS/IFA (Romania); MinES, Rosatom, RFBR and NRC “Kurchatov
Institute” (Russia); MinECo, XuntaGal and GENCAT (Spain); SNSF and SER
(Switzerland); NAS Ukraine (Ukraine); STFC (United Kingdom); NSF (USA). We
also acknowledge the support received from the ERC under FP7. The Tier1
computing centres are supported by IN2P3 (France), KIT and BMBF (Germany),
INFN (Italy), NWO and SURF (The Netherlands), PIC (Spain), GridPP (United
Kingdom). We are thankful for the computing resources put at our disposal by
Yandex LLC (Russia), as well as to the communities behind the multiple open
source software packages that we depend on.

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# Indirect Evidence for Dark Matter Density Spikes around Stellar-Mass Black
Holes

Man Ho Chan, Chak Man Lee  Department of Science and Environmental Studies,
The Education University of Hong Kong
Tai Po, New Territories, Hong Kong, China

###### Abstract

It has been suggested for a long time that dark matter would form a density
spike around a black hole. However, no promising evidence has been observed so
far to verify this theoretical suggestion. Here, we report the existence of a
dark matter density spike around each of the two nearby stellar-mass black
holes (A0620-00 and XTE J1118+480). The dynamical friction between dark matter
and the companion stars can satisfactorily explain the abnormally fast orbital
decays in the two binaries. The calculated spike index for A0620-00 and XTE
J1118+480 are $\gamma=1.71^{+0.01}_{-0.02}$ and $\gamma=1.85^{+0.04}_{-0.04}$
respectively, which are close to the lower regime predicted by the stellar
heating model. It may provide a possible indirect evidence for the existence
of dark matter density spikes around stellar-mass black holes. We anticipate
that analyzing observational data of nearby black hole X-ray binaries would be
a new way to reveal the nature of dark matter.

##  1 Introduction

Theoretical calculations have predicted that dark matter density distribution
would be altered by a massive black hole (Gondolo & Silk, 1999; Merritt, 2004;
Gnedin & Primack, 2004; Merritt, 2003; Sadeghian et al, 2013; Nampalliwar et
al, 2021). The conservation of angular momentum and energy would naturally
force dark matter to form a dense spike (i.e. a cusp-like density profile)
(Gondolo & Silk, 1999; Merritt, 2004; Gnedin & Primack, 2004; Sadeghian et al,
2013). Generally speaking, dark matter density around a black hole would
eventually follow a simple power-law form: $\rho_{\rm DM}\propto r^{-\gamma}$,
where $r$ is the radial distance from the black hole and $\gamma$ is the spike
index. The value of $\gamma$ is model-dependent, which can range from
$\gamma=1.5$ to $\gamma=2.5$ (Merritt, 2004; Gnedin & Primack, 2004; Sadeghian
et al, 2013; Fields et al, 2014; Lacroix, 2018). Since the dark matter density
profile is a singular form in $r$, the dark matter density near black hole
would be very high (i.e. a dense spike).

Based on this theoretical prediction, if dark matter can self-annihilate to
give gamma-ray photons, one can expect that the annihilation gamma-ray signals
would be greatly enhanced because the annihilation rate is proportional to
$\rho_{\rm DM}^{2}$. A lot of attention has been paid specifically for the
dark matter density spike surrounding galactic supermassive black holes
(Gondolo & Silk, 1999; Gnedin & Primack, 2004; Fields et al, 2014; Bertone et
al, 2002; Shapiro & Shelton, 2016) and intermediate-mass black holes (Lacroix,
2018; Chan, 2018). Various studies have been done to examine the possible
enhanced gamma-ray signals, especially near the supermassive black hole in the
Milky Way galaxy (Fields et al, 2014; Shapiro & Shelton, 2016). However, no
promising signals have been observed to verify the theoretical prediction
(Fields et al, 2014). However, it does not mean that the dark matter density
spike model is wrong. The negative result in gamma-ray observations could be
due to the following reasons: 1. the rest mass of dark matter particles is
very large, 2. the annihilation cross section is very small, or 3. dark matter
particles do not self-annihilate.

On the other hand, observations and studies of the two closest black hole low-
mass X-ray binaries (BH-LMXBs), A0620-00 and XTE J1118+480, have provided very
precise measurements for many important physical parameters, including the
orbital period $P$, observed radial velocity of the companion star $K$,
orbital inclination $i$, black hole mass $M_{\rm BH}$, and the mass of the
companion star $m$ (or the mass ratio $q=m/M_{\rm BH}$) (McClintock &
Remillard, 1986; Neilsen et al, 2008; Cantrell et al, 2010; Grunsven et al,
2017; Khargharia et al, 2013; Zurita, 2016; Cherepashchuk et al, 2019) (see
Table 1 for the measured values). The companion star is orbiting the black
hole in a nearly circular orbit for each binary. In particular, observations
have revealed abnormally fast orbital decays in the two BH-LMXBs:
$\dot{P}=-0.60\pm 0.08$ ms yr-1 for A0620-00 and $\dot{P}=-1.90\pm 0.57$ ms
yr-1 for XTE J1118+480 (González Hernándex et al, 2014). These decays are two
orders of magnitude larger than the one expected with gravitational wave
radiation (Chen & Li, 2015; Chen & Podsiadlowski, 2019). Standard theories
only predict $\dot{P}\sim-0.02$ ms yr-1 (González Hernández et al, 2017). Two
major proposals have been suggested recently to account for the fast orbital
decay. The first one is related to the magnetic braking of the companion star.
If the surface magnetic field of the companion star is very strong (e.g. $\geq
10^{4}$ G), the coupling between the magnetic field and the winds from the
companion star driven by X-ray irradiation from the black hole would decrease
the orbital period through tidal torques (Chen & Li, 2015; Justham et al,
2006). However, this model requires a significant mass loss from the binary
system, which has not been observed (González Hernándex et al, 2014). The
second proposal suggests that the tidal torque between the circumbinary disk
and the binary can efficiently extract the orbital angular momentum from the
binary to cause the orbital decay (Chen & Li, 2015). Nevertheless, simulations
show that the predicted mass transfer rate and the circumbinary disk mass are
much greater than the inferred values from observations (Chen & Li, 2015).
Although a few recent studies suggesting the resonant interaction between the
binary and a surrounding circumbinary disk could produce the observed orbital
period decays (Chen & Podsiadlowski, 2019; Xu & Li, 2018), the calculated
initial mass and effective temperature of the companion stars somewhat do not
match the observations (Chen & Podsiadlowski, 2019). Therefore, it is still a
mystery for the abnormally fast orbital decays in the two BH-LMXBs.

Beside the annihilation rate, the dynamical friction due to dark matter
density spike would also be very large. If a star is moving inside a
collisionless dark matter background, the star would exert a gravitational
force to pull the dark matter particles towards it. Then a concentration of
the dark matter particles would locate behind the star and exert a collective
gravitational force on the star. This collective gravitational force would
slow down the star and the resulting effect is called dynamical friction. The
idea of dynamical friction was proposed by Chandrasekhar more than 70 years
ago (Chandrasekhar, 1943). However, very surprisingly, the dynamical
frictional effect in BH-LMXBs has not been seriously examined in previous
studies. Most of the related studies are focusing on the compact binary
systems (Antonini & Merritt, 2012; Eda et al, 2015; Pani, 2015; Yue et al,
2019; Dai et al, 2022; Li et al, 2022; Becker et al, 2022; Speeney et al,
2022; Kavanagh et al, 2020). Also, no previous study has realized the possible
observable consequence of dark matter density spike surrounding a stellar-mass
black hole. In this letter, we will discuss the observed fast orbital decays
in the two closest BH-LMXBs with the idea of dynamical friction of dark matter
density spike.

##  2 The dynamical friction model

Consider a typical BH-LMXB system. The low-mass companion star with mass
$m<1M_{\odot}$ is orbiting a central black hole with mass $M_{\rm BH}$ much
greater than the stellar mass. The central black hole is almost stationary at
the center of the system. If a dark matter density spike is surrounding the
central black hole, the companion star would experience the dynamical friction
exerted by dark matter. The energy loss due to dynamical friction would
decrease the orbital period $P$ of the companion star.

The energy loss due to dynamical friction is given by (Chandrasekhar, 1943;
Yue et al, 2019):

$\dot{E}=-\frac{4\pi G^{2}\mu^{2}\rho_{\rm DM}\xi(\sigma)\ln\Lambda}{v},$ (1)

where $\mu$ is the reduced mass of the BH-LMXB,
$\ln\Lambda\approx\ln(\sqrt{M_{\rm BH}/m})$ is the Coulomb Logarithm (Kavanagh
et al, 2020), $v$ is the orbital velocity, and $\xi(\sigma)$ is a numerical
factor which depends on the distribution function and the velocity dispersion
$\sigma$ of dark matter. If we assume a Maxwell’s distribution for dark matter
and take $\sigma=200$ km/s, we will have $\xi(\sigma)\sim 0.9$. However, as
the information about dark matter is uncertain, we simply assume
$\xi(\sigma)=1$. The orbital velocity can be determined by the observed radial
velocity $K$ and the orbital inclination $i$: $v=K/\sin i$.

Using the Keplerian relation $P^{2}=4\pi^{2}a^{3}/G(M_{\rm BH}+m)$ with $a$
being the radius of the orbital motion, we can write

$\frac{\dot{P}}{P}=\frac{3\dot{a}}{2a}=-\frac{3\dot{E}}{2E},$ (2)

where $E=-GM_{\rm BH}m/2a$ is the total mechanical energy. Therefore, the
orbital decay rate can be expressed in terms of the observed parameter set {
$q$, $K$, $i$, $P$, $M_{\rm BH}$ } by:

$\dot{P}=-\frac{12\pi qGP\ln\Lambda}{(1+q)^{2}(K/\sin i)}\left[\frac{GM_{\rm
BH}(1+q)P^{2}}{4\pi^{2}}\right]^{1/3}\rho_{\rm DM},$ (3)

where $q=m/M_{\rm BH}$ is the mass ratio.

Following the dark matter density spike theory, dark matter would re-
distribute to form a density spike around the black hole in the BH-LMXB within
the spike radius $r_{\rm sp}$. We follow the standard assumption $r_{\rm
sp}=0.2r_{\rm in}$ used in many other studies (Fields et al, 2014; Eda et al,
2015), where $r_{\rm in}$ is the radius of black hole’s sphere of influence.
Outside $r_{\rm sp}$, the dark matter density would follow the local dark
matter density of their respective positions in the Milky Way. The dark matter
density around the black hole with mass $M_{\rm BH}$ can be modeled by the
following profile (Lacroix, 2018):

$\rho_{\rm DM}=\left\\{\begin{array}[]{ll}0&{\rm for}\,\,\,r\leq 2R_{s}\\\
\rho_{0}\left(\frac{r}{r_{\rm sp}}\right)^{-\gamma}&{\rm
for}\,\,\,2R_{s}<r\leq r_{\rm sp},\\\ \rho_{0}&{\rm for}\,\,\,r>r_{\rm sp}\\\
\end{array}\right.$ (4)

where $R_{s}=2GM_{\rm BH}/c^{2}$, and $\rho_{0}$ is the local dark matter
density. When the distance from the black hole is larger than the spike radius
$r_{\rm sp}$, we assume that the dark matter density would follow back to the
local dark matter density. By taking the reference value at the solar position
$\rho_{\odot}=0.33\pm 0.03$ GeV cm-3 (Ablimit et al, 2020) and following the
Navarro-Frenk-White dark matter density profile (Navarro et al, 1996), the
local dark matter densities of A0620-00 and XTE J1118+480 can be determined by
their respective positions (González Hernández et al, 2011): $\rho_{0}=0.29\pm
0.03$ GeV cm-3 for A0620-00 and $\rho_{0}=0.34\pm 0.03$ GeV cm-3 for XTE
J1118+480.

The radius of influence can be determined by (Merritt, 2003, 2004):

$M_{\rm DM}(r\leq r_{\rm in})=\int_{0}^{r_{\rm in}}4\pi r^{2}\rho_{\rm
DM}dr=2M_{\rm BH}.$ (5)

Therefore, the spike radius $r_{\rm sp}$ is also a function of $M_{\rm BH}$.
Note that the spike density profile assumed here is not an ad hoc
parametrization, but follows from theoretical calculations (Gondolo & Silk,
1999; Sadeghian et al, 2013). It is mainly determined by the black hole mass.

The spike index $\gamma$ is the only free parameter in this analysis. For a
spike of collisionless dark matter that forms about an adiabatically growing
black hole, we have $\gamma=2.25-2.5$ (Gondolo & Silk, 1999; Fields et al,
2014). However, if gravitational scattering of stars is important, the stellar
heating effect would drive the value of $\gamma$ down to a minimum value
$\gamma=1.5$ (Merritt, 2004; Gnedin & Primack, 2004). Such a change in the
spike index depends on the heating time scale, which is given by (Merritt,
2004):

$\displaystyle t_{\rm heat}$ $\displaystyle=$
$\displaystyle\frac{\sqrt{3\pi}\Gamma(0.5)M_{\rm
BH}}{18m\ln\Lambda}\left(\frac{GM_{\rm BH}}{r_{\rm
in}^{3}}\right)^{-1/2}=1.2\times 10^{15}~{}{\rm s}$ (6)
$\displaystyle\times\left(\frac{M_{\rm
BH}}{5M_{\odot}}\right)^{1/2}\left(\frac{r_{\rm in}}{5~{}\rm
pc}\right)^{3/2}\left(\frac{m}{M_{\odot}}\right)^{-1}\left(\frac{\ln\Lambda}{3}\right)^{-1},$

Here, a constant stellar density and an initial dark matter spike index
$\gamma=2.5$ are assumed (Merritt, 2004). Generally speaking, for the age of
the black hole $t_{\rm BH}\geq t_{\rm heat}$, the spike index would approach
the minimum value $\gamma=1.5$ more likely.

##  3 Results

###  3.1 Constraints on the spike index

The analytic formula gives $\dot{P}$ in terms of the precisely measured
parameters { $q$, $K$, $i$, $P$, $M_{\rm BH}$ }. We find that the typical
values of these parameters { $0.05$, $500$ km/s, $45^{\circ}$, $1$ day,
$5M_{\odot}$ } in BH-LMXBs can give $\dot{P}\sim-1$ ms yr-1 for a typical dark
matter spike density $\rho_{\rm DM}\sim 10^{-13}$ g cm-3. For our two target
BH-LMXBs, we put the corresponding measured parameters and the observed
orbital decay rates to constrain the dark matter densities at the respective
companion stellar orbits (with radius $a$): $\rho_{\rm DM}(a)\approx
7.65^{+1.62}_{-1.43}\times 10^{-13}$ g cm-3 (A0620-00) and $\rho_{\rm
DM}(a)\approx 1.60^{+1.51}_{-0.73}\times 10^{-11}$ g cm-3 (XTE J1118+480).
Following our dark matter density spike model and involving the uncertainties
of the measured parameters, we get $\gamma=1.71^{+0.01}_{-0.02}$ for A0620-00
and $\gamma=1.85^{+0.04}_{-0.04}$ for XTE J1118+480 (see Fig. 1 for the
general relation between $\dot{P}$ and $\gamma$).

As mentioned above, theoretical predictions give $1.5\leq\gamma\leq 2.5$
(Merritt, 2004; Gnedin & Primack, 2004; Fields et al, 2014; Lacroix, 2018). If
the effects of baryons or stellar heating are important, the spike index might
be close to the smallest extreme value $\gamma=1.5$ (Merritt, 2004; Gnedin &
Primack, 2004; Fields et al, 2014). In fact, this stellar heating effect is
due to the dynamical friction between stars and dark matter. Therefore, in a
BH-LMXB, the continuous gravitational scattering between the companion star
and dark matter might provide the similar stellar heating effect to reduce the
spike index to a smaller value. Nevertheless, the case for BH-LMXB is somewhat
different from the stellar heating scenario discussed in Merritt (2004);
Gnedin & Primack (2004). There is only one companion object in a BH-LMXB while
many stars are involved in the stellar heating scenario. However, recent
simulations show that the dynamical friction of the companion object with a
large mass ratio $q$ would increase the kinetic energy of the dark matter
particles in the halo and somewhat decrease the dark matter density (Kavanagh
et al, 2020), which apparently reduces the spike index. Although this is not
identical to the stellar heating scenario, both processes involve the
dynamical friction to re-distribute the dark matter density.

Using the stellar heating scenario as an analogy, we expect that the spike
index might be smaller if $t_{\rm BH}\geq t_{\rm heat}$. Using Eq. (6), the
heating time scales for A0620-00 and XTE J1118+480 are $t_{\rm heat}=3.5\times
10^{15}$ s and $t_{\rm heat}=6.1\times 10^{15}$ s respectively. Although we do
not know the ages of the black holes in the BH-LMXBs, we can assume $t_{\rm
BH}\leq P/\dot{P}$. It is because if $t_{\rm BH}>P/\dot{P}$, it should be
highly improbable for us to observe A0620-00 and XTE J1118+480 now as both
systems would have collapsed very likely within the cosmological age of 13.7
Gyr ($4.3\times 10^{17}$ s), unless a significant change of mass transfer rate
has occurred. If we follow this assumption, we can find that the upper limit
of $t_{\rm BH}$ for the A0620-00 black hole is the same order of magnitude as
the heating time scale while the upper limit of $t_{\rm BH}$ for the XTE
J1118+480 black hole is about 20 times smaller than the heating time scale
(see Table 2). This may explain why the spike index for A0620-00 is smaller.
Therefore, our results reveal a consistent picture for the dark matter spike
model and provide a very good explanation for the abnormally fast orbital
decay in the two closest BH-LMXBs. Note that our major conclusion still holds
even if the stellar heating scenario is not a good analogy.

###  3.2 The effect of dark matter annihilation

We did not assume any dark matter annihilation in the above discussion. If
dark matter annihilation rate is large enough, the central dark matter density
would approach the constant saturation density $\rho_{\rm sat}=m_{\rm
DM}/\langle\sigma v\rangle t_{\rm BH}$ (Lacroix, 2018) when $\rho_{0}(r/r_{\rm
in})^{-\gamma}>\rho_{\rm sat}$, where $m_{\rm DM}$ is the mass of a dark
matter particle and $\langle\sigma v\rangle$ is the annihilation cross
section. If the orbital decays originate from the dynamical friction of dark
matter with the saturation density (i.e. the orbital radius is smaller than
the saturation radius), we can determine the upper limits of dark matter mass
for this particular scenario. Taking the thermal annihilation cross section
$\langle\sigma v\rangle=2.2\times 10^{-26}$ cm3/s predicted by standard
cosmology (Steigman et al, 2012) and the upper limits of $t_{\rm BH}$, we can
get $m_{\rm DM}\leq 14$ GeV for A0620-00 and $m_{\rm DM}\leq 48$ GeV for XTE
J1118+480 if the companion stars are moving in the dark matter saturation
density region. In other words, if $m_{\rm DM}>48$ GeV, the companion stars in
both systems would be orbiting the corresponding black hole in the dark matter
density spike region. Since many recent stringent constraints of thermal
annihilating dark matter indicate $m_{\rm DM}\geq 100$ GeV (Ackermann et al,
2015; Chan et al, 2019; Abazajian et al, 2020; Regis et al, 2021), the dark
matter density would not be saturated at the orbital positions in A0620-00 and
XTE J1118+480.

##  4 Discussion

The existence of dark matter density spike surrounding a black hole has been
suggested for more than two decades. However, no smoking-gun evidence has been
obtained from observations. Here, we show that the effect of dynamical
friction due to dark matter density spike can satisfactorily explain the fast
orbital decay in the two closest BH-LMXBs. The resultant spike index is
$\gamma=1.7-1.8$, which is close to the value predicted by the stellar heating
model ($\gamma=1.5$) (Merritt, 2004; Gnedin & Primack, 2004). Although the BH-
LMXBs considered here are not identical to the stellar heating scenario
discussed in Gnedin & Primack (2004), recent simulations of the compact-object
inspirals show that the motion of the companion object would affect the
distribution of the dark matter density spike surrounding an intermediate-mass
black hole, especially for the mass ratio $q>10^{-3}$ (Kavanagh et al, 2020).
Therefore, we may also see similar results of the stellar heating effect in
the BH-LMXBs. Note that although the dark matter density is changing in time
during re-distribution, the dynamical friction expression used in Eq. (1) is
still applicable because the change is very slow in time (Kavanagh et al,
2020). An overall consistent picture can be described as follows. When the
black hole in a BH-LMXB is formed, the surrounding dark matter would be re-
distributed to form a density spike (probably with an initial spike index
$\gamma\approx 2-2.5$) (Gondolo & Silk, 1999). However, the dynamical friction
between dark matter and the companion star eventually help re-distribute the
dark matter density spike again to reduce the spike index to approach
$\gamma=1.7-1.8$. The orbital period is also decreasing with a fast rate $\sim
1$ ms yr-1 due to dynamical friction. If the age of the black hole is larger
than the heating time scale, the final spike index may change to a smaller
value.

We can get very small uncertainties in $\gamma$ because the uncertainties of
the measured parameters are very small, especially for A0620-00. The uncertain
factor $\xi(\sigma)$ would only change the resulting spike index slightly.
Generally speaking, our results may suggest a possible evidence of the
existence of dark matter density spike surrounding a black hole. It also
suggests that a dark matter density spike might exist around a stellar-mass
black hole ($M_{\rm BH}\sim 1-10M_{\odot}$), but not only around a
supermassive black hole (Gondolo & Silk, 1999; Merritt, 2004; Gnedin &
Primack, 2004; Lacroix et al, 2017) or an intermediate-mass black hole
(Lacroix, 2018; Dai et al, 2022; Li et al, 2022) as suggested in the past
literature. Since no previous study has focused on the case of dark matter
density spike around a stellar-mass black hole, the effect of dark matter
dynamical friction has also been neglected. In fact, one recent study has
proposed that the electron excess detected by the DAMPE experiment might
originate from the annihilating dark matter density spike in A0620-00 (Chan &
Lee, 2019). Therefore, analyzing the effect of dark matter dynamical friction
in BH-LMXBs would open a new independent way for investigating the dark matter
distribution near stellar-mass black holes.

Moreover, if dark matter annihilation effect is important so that the central
dark matter density becomes saturated, we can calculate the upper limits of
dark matter mass for this particular scenario. Since the calculated upper
limits of thermal annihilating dark matter mass $m_{\rm DM}$ are generally
smaller than the lower limits constrained from recent multi-wavelength
studies, the companion stars should be orbiting inside the dark matter density
spike rather than the saturation density. In other words, the effect of
annihilation is not important in constraining the spike index.

In fact, analyzing the effect of dynamical friction of dark matter density
spike in a binary system is not a new idea. Nevertheless, most of the related
studies have focused on the binaries of the compact objects (e.g. black hole
binaries) rather than the BH-LMXB systems (Eda et al, 2015; Pani, 2015; Yue et
al, 2019; Dai et al, 2022; Li et al, 2022; Becker et al, 2022; Speeney et al,
2022; Kavanagh et al, 2020). In compact binaries, both gravitational radiation
and dynamical friction of dark matter are significant. Therefore,
gravitational wave detection might be required to reveal the nature of dark
matter, which might contribute extra uncertainties in the constrained
parameters. Since optical and X-ray observations can give very precise
measurements for most of the important physical parameters in BH-LMXBs, we
anticipate that analyzing BH-LMXBs can better reveal the nature of the dark
matter density spike surrounding a black hole. There are at least 18 black
hole X-ray binaries in our Galaxy (Chen & Li, 2015), which can give rich
information to constrain the nature of dark matter. For example, one nearby
black hole X-ray binary Nova Muscae 1991 also shows an abnormally fast orbital
decay $\dot{P}=-20.7\pm 12.7$ ms yr-1, although the uncertainty is quite large
(González Hernández et al, 2017). Future high quality measurements may be
helpful to further confirm the existence of dark matter density spike in these
black hole X-ray binaries. This kind of analyses would open an entirely new
window for observations and theoretical studies to investigate dark matter
astrophysics (Bertone & Tait, 2018).

Table 1: The measured parameters of A0620-00 and XTE J1118+480. | A0620-00 | XTE J1118+480
---|---|---
$M_{\rm BH}$ |  $5.86\pm 0.24M_{\odot}$ (Grunsven et al, 2017) |  $7.46^{+0.34}_{-0.69}M_{\odot}$ (González Hernándex et al, 2014)
$q$ |  $0.060\pm 0.004$ (Grunsven et al, 2017) |  $0.024\pm 0.009$ (Khargharia et al, 2013)
$K$ (km/s) |  $435.4\pm 0.5$ (Neilsen et al, 2008) |  $708.8\pm 1.4$ (Khargharia et al, 2013)
$i$ |  $54^{\circ}.1\pm 1^{\circ}.1$ (Grunsven et al, 2017) |  $73^{\circ}.5\pm 5^{\circ}.5$ (Khargharia et al, 2013)
$P$ (day) |  $0.32301415(7)$ (González Hernándex et al, 2014) |  $0.16993404(5)$ (González Hernándex et al, 2014)
$\dot{P}$ (ms yr-1) |  $-0.60\pm 0.08$ (González Hernándex et al, 2014) |  $-1.90\pm 0.57$ (González Hernándex et al, 2014)
$d$ (kpc) |  $1.06\pm 0.12$ (González Hernández et al, 2011) |  $1.70\pm 0.10$ (González Hernández et al, 2011)
Table 2:  The orbital radius $a$, the calculated dark matter density at $a$, the spike index $\gamma$, the radius of influence $r_{\rm in}$, the heating time scale $t_{\rm heat}$, and the upper limit of the black hole age $t_{\rm BH}$ for each BH-LMXB based on the dark matter density spike model. | A0620-00 | XTE J1118+480
---|---|---
$a$ (AU) | $0.0169^{+0.0003}_{-0.0002}$ | $0.0118^{+0.0002}_{-0.0004}$
$\rho_{\rm DM}(a)$ (g cm-3) | $7.65^{+1.62}_{-1.43}\times 10^{-13}$ | $1.60^{+1.51}_{-0.73}\times 10^{-11}$
$\gamma$ | $1.71^{+0.01}_{-0.02}$ | $1.85^{+0.04}_{-0.04}$
$r_{\rm in}$ (pc) | $5.41^{+0.10}_{-0.09}$ | $5.34^{+0.02}_{-0.06}$
$t_{\rm heat}$ (s) | $3.5\times 10^{15}$ | $6.1\times 10^{15}$
$t_{\rm BH}$ (s) | $\leq 1.7\times 10^{15}$ | $\leq 3.5\times 10^{14}$

Figure 1: The black and red solid lines indicate the relation between $\gamma$
and $\dot{P}$ for A0620-00 and XTE J1118+480 respectively. The horizontal
dashed lines and dotted lines represent the mean values and the $1\sigma$
limits of the observed orbital decay rates (black: A0620-00; red: XTE
J1118+480).

##  5 Acknowledgements

We thank the anonymous referees for useful comments. The work described in
this paper was partially supported by a grant from the Research Grants Council
of the Hong Kong Special Administrative Region, China (Project No. EdUHK
18300922).

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