EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)

 CERN-PH-EP-2012-327 LHCb-PAPER-2012-033 Dec. 24, 2012

First observation of the decays $\kern
3.73305pt\overline{\kern-3.73305ptB}{}^{0}_{(s)}\rightarrow
D^{+}_{s}K^{-}\pi^{+}\pi^{-}$ and $\kern
3.73305pt\overline{\kern-3.73305ptB}{}^{0}_{s}\rightarrow
D_{s1}(2536)^{+}\pi^{-}$

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

The first observation of the decays $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{+}_{s}K^{-}\pi^{+}\pi^{-}$ and $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow
D^{+}_{s}K^{-}\pi^{+}\pi^{-}$ are reported using an integrated luminosity of
1.0 $\rm fb^{-1}$ recorded by the LHCb experiment. The branching fractions,
normalized with respect to $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{+}_{s}\pi^{-}\pi^{+}\pi^{-}$ and $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{+}_{s}K^{-}\pi^{+}\pi^{-}$, respectively, are measured to be

$\displaystyle{{\cal{B}}(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{+}_{s}K^{-}\pi^{+}\pi^{-})\over{\cal{B}}(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{+}_{s}\pi^{-}\pi^{+}\pi^{-})}$ $\displaystyle=(5.2\pm 0.5\pm 0.3)\times
10^{-2},$ $\displaystyle{{\cal{B}}(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow
D^{+}_{s}K^{-}\pi^{+}\pi^{-})\over{\cal{B}}(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{+}_{s}K^{-}\pi^{+}\pi^{-})}$ $\displaystyle=0.54\pm 0.07\pm 0.07,$

where the first uncertainty is statistical and the second is systematic. The
$\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{+}_{s}K^{-}\pi^{+}\pi^{-}$ decay is of particular interest as it can be
used to measure the weak phase $\gamma$. First observation of the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D_{s1}(2536)^{+}\pi^{-},~{}D_{s1}^{+}\rightarrow D^{+}_{s}\pi^{-}\pi^{+}$
decay is also presented, and its branching fraction relative to $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{+}_{s}\pi^{-}\pi^{+}\pi^{-}$ is found to be

$\displaystyle{{\cal{B}}(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D_{s1}(2536)^{+}\pi^{-},~{}D_{s1}^{+}\rightarrow
D^{+}_{s}\pi^{-}\pi^{+})\over{\cal{B}}(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{+}_{s}\pi^{-}\pi^{+}\pi^{-})}$ $\displaystyle=(4.0\pm 1.0\pm 0.4)\times
10^{-3}.$

Submitted to Physical Review D

LHCb collaboration

R. Aaij38, C. Abellan Beteta33,n, A. Adametz11, B. Adeva34, M. Adinolfi43, C.
Adrover6, A. Affolder49, Z. Ajaltouni5, J. Albrecht35, F. Alessio35, M.
Alexander48, S. Ali38, G. Alkhazov27, P. Alvarez Cartelle34, A.A. Alves Jr22,
S. Amato2, Y. Amhis36, L. Anderlini17,f, J. Anderson37, R.B. Appleby51, O.
Aquines Gutierrez10, F. Archilli18,35, A. Artamonov 32, M. Artuso53, E.
Aslanides6, G. Auriemma22,m, S. Bachmann11, J.J. Back45, C. Baesso54, W.
Baldini16, R.J. Barlow51, C. Barschel35, S. Barsuk7, W. Barter44, A. Bates48,
Th. Bauer38, A. Bay36, J. Beddow48, I. Bediaga1, S. Belogurov28, K. Belous32,
I. Belyaev28, E. Ben-Haim8, M. Benayoun8, G. Bencivenni18, S. Benson47, J.
Benton43, A. Berezhnoy29, R. Bernet37, M.-O. Bettler44, M. van Beuzekom38, A.
Bien11, S. Bifani12, T. Bird51, A. Bizzeti17,h, P.M. Bjørnstad51, T. Blake35,
F. Blanc36, C. Blanks50, J. Blouw11, S. Blusk53, A. Bobrov31, V. Bocci22, A.
Bondar31, N. Bondar27, W. Bonivento15, S. Borghi48,51, A. Borgia53, T.J.V.
Bowcock49, C. Bozzi16, T. Brambach9, J. van den Brand39, J. Bressieux36, D.
Brett51, M. Britsch10, T. Britton53, N.H. Brook43, H. Brown49, A. Büchler-
Germann37, I. Burducea26, A. Bursche37, J. Buytaert35, S. Cadeddu15, O.
Callot7, M. Calvi20,j, M. Calvo Gomez33,n, A. Camboni33, P. Campana18,35, A.
Carbone14,c, G. Carboni21,k, R. Cardinale19,i, A. Cardini15, H. Carranza-
Mejia47, L. Carson50, K. Carvalho Akiba2, G. Casse49, M. Cattaneo35, Ch.
Cauet9, M. Charles52, Ph. Charpentier35, P. Chen3,36, N. Chiapolini37, M.
Chrzaszcz 23, K. Ciba35, X. Cid Vidal34, G. Ciezarek50, P.E.L. Clarke47, M.
Clemencic35, H.V. Cliff44, J. Closier35, C. Coca26, V. Coco38, J. Cogan6, E.
Cogneras5, P. Collins35, A. Comerma-Montells33, A. Contu52,15, A. Cook43, M.
Coombes43, G. Corti35, B. Couturier35, G.A. Cowan36, D. Craik45, S.
Cunliffe50, R. Currie47, C. D’Ambrosio35, P. David8, P.N.Y. David38, I. De
Bonis4, K. De Bruyn38, S. De Capua51, M. De Cian37, J.M. De Miranda1, L. De
Paula2, P. De Simone18, D. Decamp4, M. Deckenhoff9, H. Degaudenzi36,35, L. Del
Buono8, C. Deplano15, D. Derkach14, O. Deschamps5, F. Dettori39, A. Di
Canto11, J. Dickens44, H. Dijkstra35, P. Diniz Batista1, M. Dogaru26, F.
Domingo Bonal33,n, S. Donleavy49, F. Dordei11, A. Dosil Suárez34, D.
Dossett45, A. Dovbnya40, F. Dupertuis36, R. Dzhelyadin32, A. Dziurda23, A.
Dzyuba27, S. Easo46,35, U. Egede50, V. Egorychev28, S. Eidelman31, D. van
Eijk38, S. Eisenhardt47, R. Ekelhof9, L. Eklund48, I. El Rifai5, Ch.
Elsasser37, D. Elsby42, A. Falabella14,e, C. Färber11, G. Fardell47, C.
Farinelli38, S. Farry12, V. Fave36, V. Fernandez Albor34, F. Ferreira
Rodrigues1, M. Ferro-Luzzi35, S. Filippov30, C. Fitzpatrick35, M. Fontana10,
F. Fontanelli19,i, R. Forty35, O. Francisco2, M. Frank35, C. Frei35, M.
Frosini17,f, S. Furcas20, A. Gallas Torreira34, D. Galli14,c, M. Gandelman2,
P. Gandini52, Y. Gao3, J-C. Garnier35, J. Garofoli53, P. Garosi51, J. Garra
Tico44, L. Garrido33, C. Gaspar35, R. Gauld52, E. Gersabeck11, M. Gersabeck35,
T. Gershon45,35, Ph. Ghez4, V. Gibson44, V.V. Gligorov35, C. Göbel54, D.
Golubkov28, A. Golutvin50,28,35, A. Gomes2, H. Gordon52, M. Grabalosa
Gándara33, R. Graciani Diaz33, L.A. Granado Cardoso35, E. Graugés33, G.
Graziani17, A. Grecu26, E. Greening52, S. Gregson44, O. Grünberg55, B. Gui53,
E. Gushchin30, Yu. Guz32, T. Gys35, C. Hadjivasiliou53, G. Haefeli36, C.
Haen35, S.C. Haines44, S. Hall50, T. Hampson43, S. Hansmann-Menzemer11, N.
Harnew52, S.T. Harnew43, J. Harrison51, P.F. Harrison45, T. Hartmann55, J.
He7, V. Heijne38, K. Hennessy49, P. Henrard5, J.A. Hernando Morata34, E. van
Herwijnen35, E. Hicks49, D. Hill52, M. Hoballah5, P. Hopchev4, W.
Hulsbergen38, P. Hunt52, T. Huse49, N. Hussain52, D. Hutchcroft49, D. Hynds48,
V. Iakovenko41, P. Ilten12, J. Imong43, R. Jacobsson35, A. Jaeger11, M. Jahjah
Hussein5, E. Jans38, F. Jansen38, P. Jaton36, B. Jean-Marie7, F. Jing3, M.
John52, D. Johnson52, C.R. Jones44, B. Jost35, M. Kaballo9, S. Kandybei40, M.
Karacson35, T.M. Karbach35, I.R. Kenyon42, U. Kerzel35, T. Ketel39, A.
Keune36, B. Khanji20, Y.M. Kim47, O. Kochebina7, V. Komarov36,29, R.F.
Koopman39, P. Koppenburg38, M. Korolev29, A. Kozlinskiy38, L. Kravchuk30, K.
Kreplin11, M. Kreps45, G. Krocker11, P. Krokovny31, F. Kruse9, M.
Kucharczyk20,23,j, V. Kudryavtsev31, T. Kvaratskheliya28,35, V.N. La Thi36, D.
Lacarrere35, G. Lafferty51, A. Lai15, D. Lambert47, R.W. Lambert39, E.
Lanciotti35, G. Lanfranchi18,35, C. Langenbruch35, T. Latham45, C.
Lazzeroni42, R. Le Gac6, J. van Leerdam38, J.-P. Lees4, R. Lefèvre5, A.
Leflat29,35, J. Lefrançois7, O. Leroy6, T. Lesiak23, Y. Li3, L. Li Gioi5, M.
Liles49, R. Lindner35, C. Linn11, B. Liu3, G. Liu35, J. von Loeben20, J.H.
Lopes2, E. Lopez Asamar33, N. Lopez-March36, H. Lu3, J. Luisier36, H. Luo47,
A. Mac Raighne48, F. Machefert7, I.V. Machikhiliyan4,28, F. Maciuc26, O.
Maev27,35, J. Magnin1, M. Maino20, S. Malde52, G. Manca15,d, G. Mancinelli6,
N. Mangiafave44, U. Marconi14, R. Märki36, J. Marks11, G. Martellotti22, A.
Martens8, L. Martin52, A. Martín Sánchez7, M. Martinelli38, D. Martinez
Santos35, D. Martins Tostes2, A. Massafferri1, R. Matev35, Z. Mathe35, C.
Matteuzzi20, M. Matveev27, E. Maurice6, A. Mazurov16,30,35,e, J. McCarthy42,
G. McGregor51, R. McNulty12, M. Meissner11, M. Merk38, J. Merkel9, D.A.
Milanes13, M.-N. Minard4, J. Molina Rodriguez54, S. Monteil5, D. Moran51, P.
Morawski23, R. Mountain53, I. Mous38, F. Muheim47, K. Müller37, R. Muresan26,
B. Muryn24, B. Muster36, J. Mylroie-Smith49, P. Naik43, T. Nakada36, R.
Nandakumar46, I. Nasteva1, M. Needham47, N. Neufeld35, A.D. Nguyen36, T.D.
Nguyen36, C. Nguyen-Mau36,o, M. Nicol7, V. Niess5, N. Nikitin29, T. Nikodem11,
A. Nomerotski52,35, A. Novoselov32, A. Oblakowska-Mucha24, V. Obraztsov32, S.
Oggero38, S. Ogilvy48, O. Okhrimenko41, R. Oldeman15,d,35, M. Orlandea26, J.M.
Otalora Goicochea2, P. Owen50, B.K. Pal53, A. Palano13,b, M. Palutan18, J.
Panman35, A. Papanestis46, M. Pappagallo48, C. Parkes51, C.J. Parkinson50, G.
Passaleva17, G.D. Patel49, M. Patel50, G.N. Patrick46, C. Patrignani19,i, C.
Pavel-Nicorescu26, A. Pazos Alvarez34, A. Pellegrino38, G. Penso22,l, M. Pepe
Altarelli35, S. Perazzini14,c, D.L. Perego20,j, E. Perez Trigo34, A. Pérez-
Calero Yzquierdo33, P. Perret5, M. Perrin-Terrin6, G. Pessina20, K.
Petridis50, A. Petrolini19,i, A. Phan53, E. Picatoste Olloqui33, B. Pie
Valls33, B. Pietrzyk4, T. Pilař45, D. Pinci22, S. Playfer47, M. Plo Casasus34,
F. Polci8, G. Polok23, A. Poluektov45,31, E. Polycarpo2, D. Popov10, B.
Popovici26, C. Potterat33, A. Powell52, J. Prisciandaro36, V. Pugatch41, A.
Puig Navarro36, W. Qian4, J.H. Rademacker43, B. Rakotomiaramanana36, M.S.
Rangel2, I. Raniuk40, N. Rauschmayr35, G. Raven39, S. Redford52, M.M. Reid45,
A.C. dos Reis1, S. Ricciardi46, A. Richards50, K. Rinnert49, V. Rives
Molina33, D.A. Roa Romero5, P. Robbe7, E. Rodrigues48,51, P. Rodriguez
Perez34, G.J. Rogers44, S. Roiser35, V. Romanovsky32, A. Romero Vidal34, J.
Rouvinet36, T. Ruf35, H. Ruiz33, G. Sabatino22,k, J.J. Saborido Silva34, N.
Sagidova27, P. Sail48, B. Saitta15,d, C. Salzmann37, B. Sanmartin Sedes34, M.
Sannino19,i, R. Santacesaria22, C. Santamarina Rios34, R. Santinelli35, E.
Santovetti21,k, M. Sapunov6, A. Sarti18,l, C. Satriano22,m, A. Satta21, M.
Savrie16,e, P. Schaack50, M. Schiller39, H. Schindler35, S. Schleich9, M.
Schlupp9, M. Schmelling10, B. Schmidt35, O. Schneider36, A. Schopper35, M.-H.
Schune7, R. Schwemmer35, B. Sciascia18, A. Sciubba18,l, M. Seco34, A.
Semennikov28, K. Senderowska24, I. Sepp50, N. Serra37, J. Serrano6, P.
Seyfert11, M. Shapkin32, I. Shapoval40,35, P. Shatalov28, Y. Shcheglov27, T.
Shears49,35, L. Shekhtman31, O. Shevchenko40, V. Shevchenko28, A. Shires50, R.
Silva Coutinho45, T. Skwarnicki53, N.A. Smith49, E. Smith52,46, M. Smith51, K.
Sobczak5, F.J.P. Soler48, F. Soomro18,35, D. Souza43, B. Souza De Paula2, B.
Spaan9, A. Sparkes47, P. Spradlin48, F. Stagni35, S. Stahl11, O. Steinkamp37,
S. Stoica26, S. Stone53, B. Storaci38, M. Straticiuc26, U. Straumann37, V.K.
Subbiah35, S. Swientek9, M. Szczekowski25, P. Szczypka36,35, T. Szumlak24, S.
T’Jampens4, M. Teklishyn7, E. Teodorescu26, F. Teubert35, C. Thomas52, E.
Thomas35, J. van Tilburg11, V. Tisserand4, M. Tobin37, S. Tolk39, D.
Tonelli35, S. Topp-Joergensen52, N. Torr52, E. Tournefier4,50, S. Tourneur36,
M.T. Tran36, A. Tsaregorodtsev6, P. Tsopelas38, N. Tuning38, M. Ubeda
Garcia35, A. Ukleja25, D. Urner51, U. Uwer11, V. Vagnoni14, G. Valenti14, R.
Vazquez Gomez33, P. Vazquez Regueiro34, S. Vecchi16, J.J. Velthuis43, M.
Veltri17,g, G. Veneziano36, M. Vesterinen35, B. Viaud7, I. Videau7, D.
Vieira2, X. Vilasis-Cardona33,n, J. Visniakov34, A. Vollhardt37, D.
Volyanskyy10, D. Voong43, A. Vorobyev27, V. Vorobyev31, C. Voß55, H. Voss10,
R. Waldi55, R. Wallace12, S. Wandernoth11, J. Wang53, D.R. Ward44, N.K.
Watson42, A.D. Webber51, D. Websdale50, M. Whitehead45, J. Wicht35, D.
Wiedner11, L. Wiggers38, G. Wilkinson52, M.P. Williams45,46, M. Williams50,p,
F.F. Wilson46, J. Wishahi9, M. Witek23, W. Witzeling35, S.A. Wotton44, S.
Wright44, S. Wu3, K. Wyllie35, Y. Xie47,35, F. Xing52, Z. Xing53, Z. Yang3, R.
Young47, X. Yuan3, O. Yushchenko32, M. Zangoli14, M. Zavertyaev10,a, F.
Zhang3, L. Zhang53, W.C. Zhang12, Y. Zhang3, A. Zhelezov11, L. Zhong3, A.
Zvyagin35.

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

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

##  1 Introduction

In the Standard Model (SM), the amplitudes associated with flavor-changing
processes depend on four Cabibbo-Kobayashi-Maskawa (CKM) [1, 2] matrix
parameters. Contributions from physics beyond the Standard Model (BSM) add
coherently to these amplitudes, leading to potential deviations in rates and
CP-violating asymmetries when compared to the SM contributions alone. Since
the SM does not predict the CKM parameters, it is important to make precise
measurements of their values in processes that are expected to be insensitive
to BSM contributions. Their values then provide a benchmark to which BSM-
sensitive measurements can be compared.

The least well-determined of the CKM parameters is the weak phase
${\gamma\equiv{\rm arg}\left(-{V_{\rm ub}^{*}V_{\rm ud}\over V_{\rm
cb}^{*}V_{\rm cd}}\right)}$, which, through direct measurements, is known to a
precision of ${\sim 10^{\rm o}-12^{\rm o}}$ [3, 4]. It may be probed using
time-independent rates of decays such as $B^{-}\rightarrow DK^{-}$ [5,
*Dunietz:1992ti, *Atwood:1994zm, *Atwood:1996ci, 9, *Gronau:1991dp, 11], or by
analyzing the time-dependent decay rates of processes such as
$B_{s}^{0}\rightarrow D_{s}^{\mp}K^{\pm}$ [12, 13, 14, 15]. Sensitivity to the
weak phase $\gamma$ results from the interference between $b\rightarrow c$ and
$b\rightarrow u$ transitions, as indicated in Figs. 1(a-c). Such measurements
may be extended to multibody decay modes, such as $B^{-}\rightarrow
DK^{-}\pi^{+}\pi^{-}$ [16] for a time-independent measurement, or $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{+}_{s}K^{-}\pi^{+}\pi^{-}$ in the case of a time-dependent analysis.

The $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow
D^{+}K^{-}\pi^{+}\pi^{-}$ decay, while having the same final state as $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{+}_{s}K^{-}\pi^{+}\pi^{-}$, receives contributions not only from the
$W$-exchange process (Fig. 1(d)), but also from $b\rightarrow c$ transitions
in association with the production of an extra $s\bar{s}$ pair (Figs. 1(e-f)).
The decay may also proceed through mixing followed by a $b\rightarrow u$,
$W$-exchange process (not shown). However, this amplitude is Cabibbo-,
helicity- and color-suppressed, and is therefore negligible compared to the
$b\rightarrow c$ amplitude.

Figure 1: Diagrams contributing to the $B^{0}_{s},\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{+}_{s}K^{-}\pi^{+}\pi^{-}$ (a-c) and $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow
D^{+}_{s}K^{-}\pi^{+}\pi^{-}$ (d-f) decays, as described in the text. In
(a-d), the additional ($\pi^{+}\pi^{-}$) indicates that the
$K^{-}\pi^{+}\pi^{-}$ may be produced either through an excited strange kaon
resonance decay, or through fragmentation.

This paper reports the first observation of $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{+}_{s}K^{-}\pi^{+}\pi^{-}$ and $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow
D^{+}_{s}K^{-}\pi^{+}\pi^{-}$, and measurements of their branching fractions
relative to $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{+}_{s}\pi^{-}\pi^{+}\pi^{-}$ and $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{+}_{s}K^{-}\pi^{+}\pi^{-}$, respectively. The data sample is based on an
integrated luminosity of 1.0 $\rm fb^{-1}$ of $pp$ collisions at $\sqrt{s}=7$
$\mathrm{\,Te\kern-1.00006ptV}$ collected by the LHCb experiment. The same
data sample is also used to observe the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D_{s1}(2536)^{+}\pi^{-},~{}D_{s1}^{+}\rightarrow D^{+}_{s}\pi^{-}\pi^{+}$
decay for the first time, and measure its branching fraction relative to
$\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{+}_{s}\pi^{-}\pi^{+}\pi^{-}$. The inclusion of charge-conjugated modes is
implied throughout this paper.

##  2 Detector and simulation

The LHCb detector [17] 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 has a momentum resolution ($\Delta p/p$) 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 (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. Photon, electron and hadron candidates are identified by a
calorimeter system consisting of scintillating-pad and pre-shower detectors,
an electromagnetic calorimeter and a hadronic calorimeter. Muons are
identified by a system composed of alternating layers of iron and multiwire
proportional chambers.

The trigger consists of a hardware stage, based on information from the
calorimeter and muon systems, followed by a software stage, which applies a
full event reconstruction. The software trigger requires a two-, three- or
four-track secondary vertex with a high $p_{\rm T}$ sum of the tracks and a
significant displacement from the primary $pp$ interaction vertices (PVs). At
least one track should have $\mbox{$p_{\rm
T}$}>1.7{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$, an IP $\chi^{2}$ greater than
16 with respect to all PVs, and a track fit $\chi^{2}/\rm{ndf}<2$, where ndf
is the number of degrees of freedom. The IP $\chi^{2}$ is defined as the
difference between the $\chi^{2}$ of the PV reconstructed with and without the
considered particle. A multivariate algorithm is used for the identification
of secondary vertices [18].

For 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].

##  3 Signal selection

Signal $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{(s)}$ decay
candidates are formed by pairing a $D^{+}_{s}\rightarrow K^{+}K^{-}\pi^{+}$
candidate with either a $\pi^{-}\pi^{+}\pi^{-}$ (hereafter referred to as
$X_{d}$) or a $K^{-}\pi^{+}\pi^{-}$ combination (hereafter referred to as
$X_{s}$). Tracks used to form the $D^{+}_{s}$ and $X_{d,s}$ are required to be
identified as either a pion or a kaon using information from the RICH
detectors, have $p_{\rm T}$ in excess of 100
${\mathrm{\,Me\kern-1.00006ptV\\!/}c}$, and be significantly detached from any
reconstructed PV in the event.

Signal $D^{+}_{s}$ candidates are required to have good vertex fit quality, be
significantly displaced from the nearest PV, and have invariant mass,
$M(K^{+}K^{-}\pi^{+})$, within 20 ${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$
of the $D^{+}_{s}$ mass [26]. To suppress combinatorial and charmless
backgrounds, only those $D^{+}_{s}$ candidates that are consistent with
decaying through either the $\phi$
($M(K^{+}K^{-})<1040~{}{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$) or $\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$
($|M(K^{-}\pi^{+})-m_{K^{*0}}|<75~{}{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$)
resonances are used (here, $m_{K^{*0}}$ is the $K^{*0}$ mass [26]). The
remaining charmless background yields are determined using the $D^{+}_{s}$
mass sidebands. For about 20% of candidates, when the $K^{+}$ is assumed to be
a $\pi^{+}$, the corresponding $K^{-}\pi^{+}\pi^{+}$ invariant mass is
consistent with the $D^{+}$ mass. To suppress cross-feed from $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow D^{+}X$ decays, a
tighter particle identification (PID) requirement is applied to the $K^{+}$ in
the $D^{+}_{s}\rightarrow K^{+}K^{-}\pi^{+}$ candidates when
$|M(K^{-}\pi^{+}\pi^{+})-m_{D^{+}}|<20~{}{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$
($m_{D^{+}}$ is the $D^{+}$ mass [26]). Similarly, if the invariant mass of
the particles forming the $D^{+}_{s}$ candidate, after replacing the $K^{+}$
mass with the proton mass, falls within 15
${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ of the $\Lambda_{c}^{+}$ mass,
tighter PID selection is applied. The sizes of these mass windows are about
2.5 times the invariant mass resolution, and are sufficient to render these
cross-feed backgrounds negligible.

Candidate $X_{d}$ and $X_{s}$ are formed from $\pi^{-}\pi^{+}\pi^{-}$ or
$K^{-}\pi^{+}\pi^{-}$ combinations, where all invariant mass values up to 3
${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$ are accepted. To reduce the level
of combinatorial background, we demand that the $X_{d,s}$ vertex is displaced
from the nearest PV by more than 100 $\,\upmu\rm m$ in the direction
transverse to the beam axis and that at least two of the daughter tracks have
$\mbox{$p_{\rm T}$}>300~{}{\mathrm{\,Me\kern-1.00006ptV\\!/}c}$. Backgrounds
to the $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{(s)}\rightarrow
D^{+}_{s}K^{-}\pi^{+}\pi^{-}$ search from $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D_{s}^{(*)+}\pi^{-}\pi^{+}\pi^{-}$ or $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{+}_{s}K^{-}K^{+}\pi^{-}$ decays are suppressed by applying more stringent
PID requirements to the $K^{-}$ and $\pi^{+}$ in $X_{s}$. The PID requirements
have an efficiency of about 65% for selecting $X_{s}$, while rejecting about
97% of the favored three-pion background. To suppress peaking backgrounds from
$\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{+}_{s}D^{-}_{s}$ decays, where
$D^{+}_{s}\rightarrow\pi^{+}\pi^{-}\pi^{+},~{}K^{+}\pi^{-}\pi^{+}$, it is
required that $M(X_{d,s})$ is more than 20
${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ away from the $D^{+}_{s}$ mass.

Signal $\kern 1.79993pt\overline{\kern-1.79993ptB}{}$ meson candidates are
then formed by combining a $D^{+}_{s}$ with either an $X_{d}$ or $X_{s}$. The
reconstructed $\kern 1.79993pt\overline{\kern-1.79993ptB}{}$ candidate is
required to be well separated from the nearest PV with a decay time larger
than 0.2 ps and have a good quality vertex fit. To suppress remaining
charmless backgrounds, which appear primarily in $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow
D^{+}_{s}K^{-}\pi^{+}\pi^{-}$, the vertex separation (VS) $\chi^{2}$ between
the $D^{+}_{s}$ and $\kern 1.79993pt\overline{\kern-1.79993ptB}{}$ decay
vertices is required to be greater than 9. Candidates passing all selection
requirements are refit with both $D^{+}_{s}$ mass and vertex constraints to
improve the mass resolution [27].

To further suppress combinatorial background, a boosted decision tree (BDT)
selection [28, *Roe] with the AdaBoost algorithm[30] is employed. The BDT is
trained using simulated $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{+}_{s}K^{-}\pi^{+}\pi^{-}$ decays for the signal distributions, and the
high $\kern 1.79993pt\overline{\kern-1.79993ptB}{}$ mass sideband in data are
used to model the backgrounds. The following thirteen variables are used:

  * •

$\kern 1.79993pt\overline{\kern-1.79993ptB}{}$ candidate: IP $\chi^{2}$, VS
$\chi^{2}$, vertex fit $\chi^{2}$, and $p_{\rm T}$;

  * •

$D^{+}_{s}$ candidate: Flight distance significance from $\kern
1.79993pt\overline{\kern-1.79993ptB}{}$ vertex;

  * •

$X_{d,s}$ candidate: IP $\chi^{2}$, maximum of the distances of closest
approach between any pair of tracks in the decay;

  * •

$X_{d,s}$ daughters: min(IP $\chi^{2}$), max(IP $\chi^{2}$), min($p_{\rm T}$);

  * •

$D^{+}_{s}$ daughters: min(IP $\chi^{2}$), max(IP $\chi^{2}$), min($p_{\rm
T}$),

where min and max denote the minimum and maximum of the indicated values
amongst the daughter particles. The flight distance significance is the
separation between the $D^{+}_{s}$ and $\kern
1.79993pt\overline{\kern-1.79993ptB}{}$ vertices, normalized by the
uncertainty. The training produces a single variable, $x$, that provides
discrimination between signal decays and background contributions. The cut
value is chosen by optimizing $S(x_{\rm cut})/\sqrt{S(x_{\rm cut})+B(x_{\rm
cut})}$, where $S(x_{\rm cut})$ and $B(x_{\rm cut})$ are the expected signal
and background yields, respectively, after requiring $x>x_{\rm cut}$. At the
optimal point, a signal efficiency of $\sim$90% is expected while rejecting
about $85\%$ of the combinatorial background (after the previously discussed
selections are applied). After all selections, about 3% of events have more
than one signal candidate in both data and simulation. All candidates are kept
for further analysis.

##  4 Fits to data

The $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{+}_{s}\pi^{-}\pi^{+}\pi^{-}$ and $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{(s)}\rightarrow
D^{+}_{s}K^{-}\pi^{+}\pi^{-}$ invariant mass spectra are each modeled by the
sum of a signal and several background components. The signal shapes are
obtained from simulation, and are each described by the sum of a Crystal Ball
(CB) [31] shape and a Gaussian function. The CB shape parameter that describes
the tail toward low mass is fixed based on simulated decays. A common, freely
varying scale factor multiplies the width parameters in the CB and Gaussian
functions to account for slightly larger resolution in data than in
simulation. For the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{(s)}\rightarrow
D^{+}_{s}K^{-}\pi^{+}\pi^{-}$ mass fit, the difference between the mean $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ and $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ masses is fixed to 87.35
${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ [26].

Several non-signal $b$-hadron decays produce broad peaking structures in the
$D^{+}_{s}\pi^{-}\pi^{+}\pi^{-}$ and $D^{+}_{s}K^{-}\pi^{+}\pi^{-}$ invariant
mass spectra. For $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{+}_{s}\pi^{-}\pi^{+}\pi^{-}$, the only significant source of peaking
background is from $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{*+}_{s}\pi^{-}\pi^{+}\pi^{-}$, where the photon or $\pi^{0}$ from the
$D^{*+}_{s}$ decay is not included in the reconstructed decay. Since the full
decay amplitude for $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{*+}_{s}\pi^{-}\pi^{+}\pi^{-}$ is not known, the simulation may not
adequately model the decay. Simulation is therefore used to provide an
estimate for the shape, but the parameters are allowed to vary within one
standard deviation about the fitted values.

For $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{(s)}\rightarrow
D^{+}_{s}K^{-}\pi^{+}\pi^{-}$, backgrounds from $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{(s)}\rightarrow
D^{*+}_{s}K^{-}\pi^{+}\pi^{-}$ and from misidentified $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{+}_{s}\pi^{-}\pi^{+}\pi^{-}$ and $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{*+}_{s}\pi^{-}\pi^{+}\pi^{-}$ decays are considered. The $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{(s)}\rightarrow
D^{*+}_{s}K^{-}\pi^{+}\pi^{-}$ shape is fixed to be the same as that obtained
for the $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{*+}_{s}\pi^{-}\pi^{+}\pi^{-}$ component in the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{+}_{s}\pi^{-}\pi^{+}\pi^{-}$ mass fit. This same shape is assumed for both
$\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ and $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$, where for the former, a shift
by the $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}-\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ mass difference is included.
For the $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{+}_{s}\pi^{-}\pi^{+}\pi^{-}$ and $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{*+}_{s}\pi^{-}\pi^{+}\pi^{-}$ cross-feed, simulated decays and kaon
misidentification rates taken from $D^{*+}$ calibration data are used to
obtain their expected yields and invariant mass shapes. The cross-feed
contribution is about 3% of the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{+}_{s}\pi^{-}\pi^{+}\pi^{-}$ and $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{*+}_{s}\pi^{-}\pi^{+}\pi^{-}$ yields; the corresponding cross-feed yields
are fixed in the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{(s)}\rightarrow
D^{+}_{s}K^{-}\pi^{+}\pi^{-}$ fit. The shape is obtained by parameterizing the
invariant mass spectrum obtained from the simulation after replacing the
appropriate $\pi^{-}$ mass in $X_{d}$ with the kaon mass. The combinatorial
background is described by an exponential function whose slope is allowed to
vary independently for both mass fits.

Figure 2 shows the invariant mass distribution for $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{+}_{s}\pi^{-}\pi^{+}\pi^{-}$ candidates passing all selection criteria. The
fitted number of $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{+}_{s}\pi^{-}\pi^{+}\pi^{-}$ signal events is $5683\pm 83$. While it is
expected that most of the low mass background emanates from $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{*+}_{s}\pi^{-}\pi^{+}\pi^{-}$ decays, contributions from other sources such
as $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{+}_{s}\pi^{-}\pi^{+}\pi^{-}\pi^{0}$ are also possibly absorbed into this
background component. Figure 3 shows the invariant mass distribution for
$\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{(s)}\rightarrow
D^{+}_{s}K^{-}\pi^{+}\pi^{-}$ candidates. The fitted signal yields are $402\pm
33$ $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow
D^{+}_{s}K^{-}\pi^{+}\pi^{-}$ and $216\pm 21$ $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{+}_{s}K^{-}\pi^{+}\pi^{-}$ events.

Figure 2: Invariant mass distribution for $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{+}_{s}\pi^{-}\pi^{+}\pi^{-}$ candidates. The fitted signal probability
disrtibution function (PDF) is indicated by the dashed line and the background
shapes are shown as shaded regions, as described in the text. Figure 3:
Invariant mass distribution for $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{(s)}\rightarrow
D^{+}_{s}K^{-}\pi^{+}\pi^{-}$ candidates. The fitted signal (dashed lines) and
background shapes (shaded/hatched regions) are shown, as described in the
text.

The $D^{+}_{s}$ mass sidebands, defined to be from 35 to 55
${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ on either side of the nominal
$D^{+}_{s}$ mass, are used to estimate the residual charmless background that
may contribute to the observed signals. The numbers of $B^{0}_{s}$ decays in
the $D^{+}_{s}$ sidebands are $61\pm 16$ , $0^{+5}_{-0}$, and $9\pm 5$ for the
$\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{+}_{s}\pi^{-}\pi^{+}\pi^{-}$, $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{+}_{s}K^{-}\pi^{+}\pi^{-}$ and $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow
D^{+}_{s}K^{-}\pi^{+}\pi^{-}$ decays, respectively; they are subtracted from
the observed signal yields to obtain the corrected number of signal decays.
The yields in the signal and sideband regions are summarized in Table 1.

Table 1: Summary of event yields from data in the $D^{+}_{s}$ signal and sidebands regions, and the background corrected yield. The signal and sideband regions require $D^{+}_{s}$ candidates to have invariant mass $|M(K^{+}K^{-}\pi^{+})-m_{D^{+}_{s}}|<20$ ${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ and $35<|M(K^{+}K^{-}\pi^{+})-m_{D^{+}_{s}}|<55$ ${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$, respectively, where $m_{D^{+}_{s}}$ is the $D^{+}_{s}$ mass [26]. Decay | Signal Region | Sideband Region | Corrected Yield
---|---|---|---
$\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow D^{+}_{s}\pi^{-}\pi^{+}\pi^{-}$ | $5683\pm 83$ | $\phantom{1}61\pm 16$ | $5622\pm 85$
$\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow D^{+}_{s}K^{-}\pi^{+}\pi^{-}$ | $\phantom{1}216\pm 21$ | $\phantom{1}0^{+5}_{-0}$ | $\phantom{1}216\pm 22$
$\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow D^{+}_{s}K^{-}\pi^{+}\pi^{-}$ | $\phantom{1}402\pm 33$ | $~{}~{}9\pm 5$ | $\phantom{1}393\pm 33$

##  5 Mass distributions of ${\mathbf{X}_{\boldsymbol{d,s}}}$ and two-body
masses

In order to investigate the properties of these $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{(s)}$ decays, sWeights [32]
obtained from the mass fits are used to determine the underlying $X_{d,s}$
invariant mass spectra as well as the two-body invariant masses amongst the
three daughter particles. Figure 4 shows (a) the $\pi^{-}\pi^{+}\pi^{-}$ mass,
(b) the smaller $\pi^{+}\pi^{-}$ mass and (c) the larger $\pi^{+}\pi^{-}$ mass
in $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{+}_{s}\pi^{-}\pi^{+}\pi^{-}$ data and simulated decays. A prominent peak,
consistent with the $a_{1}(1260)^{-}\rightarrow\pi^{-}\pi^{+}\pi^{-}$ is
observed, along with structures consistent with the $\rho^{0}$ in the two-body
masses. There appears to be an offset in the peak position of the
$a_{1}(1260)^{-}$ between data and simulation. Since the mean and width of the
$a_{1}(1260)^{-}$ resonance are not well known, and their values may even be
process dependent, this level of agreement is reasonable. A number of other
spectra have been compared between data and simulation, such as the $p_{\rm
T}$ spectra of the $D^{+}_{s}$, $X_{d}$ and the daughter particles, and
excellent agreement is found.

Figure 4: Invariant mass distributions for (a) $X_{d}$, (b) smaller
$\pi^{+}\pi^{-}$ mass in $X_{d}$ and (c) the larger $\pi^{+}\pi^{-}$ mass in
$X_{d}$, from $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{+}_{s}\pi^{-}\pi^{+}\pi^{-}$ decays using sWeights. The points are the data
and the solid line is the simulation. The simulated distribution is normalized
to have the same yield as the data.

Figure 5 shows the corresponding distributions for the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{+}_{s}K^{-}\pi^{+}\pi^{-}$ decay. A peaked structure at low
$K^{-}\pi^{+}\pi^{-}$ mass, consistent with contributions from the lower-lying
excited strange mesons, such as the $K_{1}(1270)^{-}$ and $K_{1}(1400)^{-}$,
is observed. As many of these states decay through $\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$ and $\rho^{0}$ mesons,
significant contributions from these resonances are observed in the
$K^{-}\pi^{+}$ and $\pi^{+}\pi^{-}$ invariant mass spectra, respectively. The
simulation provides a reasonable description of the distributions in the data.

Figure 5: Invariant mass distributions for (a) $X_{s}$, (b) $\pi^{+}\pi^{-}$
in $X_{s}$ and (c) the $K^{-}\pi^{+}$ in $X_{s}$, from $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{+}_{s}K^{-}\pi^{+}\pi^{-}$ data using sWeights. The points are data and the
solid line is the simulation. The simulated distribution is normalized to have
the same yield as the data.

Figure 6 shows the same distributions for $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow
D^{+}_{s}K^{-}\pi^{+}\pi^{-}$. The $K^{-}\pi^{+}\pi^{-}$ invariant mass is
quite broad, with little indication of any narrow structures. There are
indications of $\kern 1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$ and
$\rho^{0}$ contributions in the $K^{-}\pi^{+}$ and $\pi^{-}\pi^{+}$ invariant
mass spectra, respectively, but the contribution from resonances such as the
$K_{1}(1270)^{-}$ or $K_{1}(1400)^{-}$ appear to be small or absent. In the
$K^{-}\pi^{+}$ invariant mass spectrum, there may be an indication of a $\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*}_{0}(1430)^{0}$ contribution. The
simulation, which models the $K^{-}\pi^{+}\pi^{-}$ final state as 10%
$K_{1}(1270)^{-}$, 10% $K_{1}(1400)^{-}$, 40% $\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}\pi^{-}$ and 40% $K^{-}\rho^{0}$,
provides a reasonable description of the data, which suggests that processes
such as those in Figs. 1(e-f) constitute a large portion of the total width
for this decay.

Figure 6: Invariant mass distributions for (a) $X_{s}$, (b) $\pi^{+}\pi^{-}$
in $X_{s}$ and (c) the $K^{-}\pi^{+}$ in $X_{s}$, from $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow
D^{+}_{s}K^{-}\pi^{+}\pi^{-}$ data using sWeights. The points are data and the
solid line is the simulation. The simulated distribution is normalized to have
the same yield as the data.

##  6 First observation of $\boldsymbol{\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow~{}D_{s1}(2536)^{+}\pi^{-}}$

A search for excited $D^{+}_{s}$ states, such as $D_{sJ}^{+}\rightarrow
D^{+}_{s}\pi^{-}\pi^{+}$, contributing to the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{+}_{s}\pi^{-}\pi^{+}\pi^{-}$ final state is performed. Signal candidates
within $\pm$40 ${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ of the nominal
$B^{0}_{s}$ mass are selected, and from them the invariant mass difference,
$\Delta M=M(D^{+}_{s}\pi^{-}\pi^{+})-M(D^{+}_{s}$) is formed, where both
$\pi^{+}\pi^{-}$ combinations are included. The $\Delta M$ distribution for
candidates in the $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$
signal window is shown in Fig. 7. A peak corresponding to the
$D_{s1}(2536)^{+}$ is observed, whereas no significant structures are observed
in the upper $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ mass
sideband (5450$-$5590 ${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$). The
distribution is fitted to the sum of a signal Breit-Wigner shape convolved
with a Gaussian resolution function, and a second order polynomial to describe
the background contribution. The Breit-Wigner width is set to 0.92
${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ [26], and the Gaussian resolution
is fixed to 3.8 ${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ based on
simulation. A signal yield of $20.0\pm 5.1$ signal events is observed at a
mass difference of $565.1\pm 1.0~{}{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$,
which is consistent with the known $D_{s1}(2536)^{+}-D^{+}_{s}$ mass
difference of $566.63\pm 0.35~{}{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$
[26]. The significance of the signal is 5.9, obtained by fitting the invariant
mass distribution with the mean mass difference fixed to 566.63
${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ [26], and computing $\sqrt{-2{\rm
ln}({\mathcal{L}_{0}}/{\mathcal{L}_{\rm max}}})$. Here, ${\mathcal{L}_{\rm
max}}$ and ${\mathcal{L}_{0}}$ are the fit likelihoods with the signal yields
left free and fixed to zero, respectively. Several variations in the
background shape were investigated, and in all cases the signal significance
exceeded 5.5. This decay is therefore observed for the first time. To obtain
the yield in the normalization mode ($\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{+}_{s}\pi^{-}\pi^{+}\pi^{-}$), the signal function is integrated from 40
${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ below to 40
${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ above the nominal $B^{0}_{s}$ mass.
A yield of $5505\pm 85$ events is found in this restricted mass interval.

Figure 7: Distribution of the difference in invariant mass,
$M(D^{+}_{s}\pi^{-}\pi^{+})-M(D^{+}_{s})$, using $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{+}_{s}\pi^{-}\pi^{+}\pi^{-}$ candidates within 40
${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ of the known $B^{0}_{s}$ mass
(points) and in the upper $B^{0}_{s}$ mass sidebands (filled histogram). The
fit to the distribution is shown, as described in the text.

##  7 Selection efficiencies

The ratios of branching fractions can be written as

$\displaystyle{{\cal{B}}(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{+}_{s}K^{-}\pi^{+}\pi^{-})\over{\cal{B}}(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{+}_{s}\pi^{-}\pi^{+}\pi^{-})}$ $\displaystyle={Y(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{+}_{s}K^{-}\pi^{+}\pi^{-})\over Y(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{+}_{s}\pi^{-}\pi^{+}\pi^{-})}\times\epsilon_{\rm rel}^{s}$ (1)

and

$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}{{\cal{B}}(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow
D^{+}_{s}K^{-}\pi^{+}\pi^{-})\over{\cal{B}}(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{+}_{s}K^{-}\pi^{+}\pi^{-})}$ $\displaystyle={Y(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow
D^{+}_{s}K^{-}\pi^{+}\pi^{-})\over Y(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{+}_{s}K^{-}\pi^{+}\pi^{-})}\times\epsilon_{\rm rel}^{d}\times f_{s}/f_{d},$
(2)

where $Y$ are the measured yields, $\epsilon_{\rm rel}^{s}=\epsilon(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{+}_{s}\pi^{-}\pi^{+}\pi^{-})/\epsilon(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{+}_{s}K^{-}\pi^{+}\pi^{-})$ and $\epsilon_{\rm rel}^{d}=\epsilon(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{+}_{s}K^{-}\pi^{+}\pi^{-}))/\epsilon(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow
D^{+}_{s}K^{-}\pi^{+}\pi^{-})$ are the relative selection efficiencies
(including trigger), and $f_{s}/f_{d}=0.267\pm 0.021$ [33] is the $B^{0}_{s}$
fragmentation fraction relative to $B^{0}$. The ratios of selection
efficiencies are obtained from simulation, except for the PID requirements,
which are obtained from a dedicated $D^{*+}$ calibration sample, weighted to
match the momentum spectrum of the particles that form $X_{d}$ and $X_{s}$.
The selection efficiencies for each decay are given in Table 2. The efficiency
of the $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{+}_{s}\pi^{-}\pi^{+}\pi^{-}$ decay is about 35% larger than the values
obtained in either the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{+}_{s}K^{-}\pi^{+}\pi^{-}$ or $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow
D^{+}_{s}K^{-}\pi^{+}\pi^{-}$ decay; the efficiencies of the latter two are
consistent with each other. The lower efficiency is due almost entirely to the
tighter PID requirements on the $K^{-}$ and $\pi^{+}$ in $X_{s}$. Two
additional multiplicative correction factors, also shown in Table 2, are
applied to the measured ratio of branching fractions in Eqs. 1 and 2. The
first is a correction for the $D^{+}_{s}$ mass veto on $M(X_{d,s})$, and the
second is due to the requirement that
$M(X_{s,d})<3~{}{\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$. The former, which
represents a small correction, is estimated from the sWeight-ed distributions
of $M(X_{d,s})$ shown previously. For the latter, the fraction of events with
$M(X_{d,s})>3~{}{\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$ is obtained from
simulation, and scaled by the ratio of yields in data relative to simulation
for the mass region
$2.6<M(X_{s,d})<3.0~{}{\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$. A 50%
uncertainty is assigned to the estimated correction. Based on the qualitative
agreement between data and simulation in the $M(X_{d,s})$ distributions (see
Sect. 5), and the fact that the phase space approaches zero as
$M(X_{d,s})\rightarrow~{}3.5~{}{\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$, this
uncertainty is conservative.

Table 2: Selection efficiencies and correction factors for decay modes under study. The uncertainties on the selection efficiencies are statistical only, whereas the correction factors show the total uncertainty. Quantity | $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow D^{+}_{s}\pi^{-}\pi^{+}\pi^{-}$ | $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow D^{+}_{s}K^{-}\pi^{+}\pi^{-}$ | $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow D^{+}_{s}K^{-}\pi^{+}\pi^{-}$
---|---|---|---
Total $\epsilon$ ($10^{-4}$) | $4.97\pm 0.08$ | $3.67\pm 0.10$ | $3.59\pm 0.10$
$D^{+}_{s}$ veto corr. | $1.013\pm 0.003$ | $1.013\pm 0.003$ | $1.017\pm 0.005$
$M>3~{}{\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$ corr. | $1.02\pm 0.01$ | $1.04\pm 0.02$ | $1.14\pm 0.07$

The relative efficiency between $B^{0}_{s}\rightarrow
D_{s1}(2536)^{+}\pi^{-},~{}D_{s1}^{+}\rightarrow D^{+}_{s}\pi^{-}\pi^{+}$ and
$\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{+}_{s}\pi^{-}\pi^{+}\pi^{-}$ is estimated from simulation, and is found to
be $0.90\pm 0.05$.

##  8 Systematic uncertainties

Several uncertainties contribute to the ratio of branching fractions. The
sources and their values are listed in Table 3. The largest uncertainty, which
applies only to the ratio ${{\cal{B}}(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow
D^{+}_{s}K^{-}\pi^{+}\pi^{-})\over{\cal{B}}(\kern
1.25995pt\overline{\kern-1.25995ptB}{}^{0}_{s}\rightarrow
D^{+}_{s}K^{-}\pi^{+}\pi^{-})}$, is from the $b$ hadronization fraction,
$f_{s}/f_{d}=0.267\pm 0.021$ [33], which is 7.9%. Another large uncertainty
results from the required correction factor to account for the signal with
$M(X_{s,d})>3~{}{\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$. Those corrections
are described in Sect. 7.

The selection efficiency depends slightly on the modeling of the $X_{d,s}$
decay. The momentum spectra of the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}$, $D^{+}_{s}$, $X_{d,s}$ and the
$X_{d,s}$ daughters have been compared to simulation, and excellent agreement
is found. The selection efficiency is consistent with being flat as a function
of $M(X_{d,s})$ at the level of two standard deviations or less. To assess a
potential systematic uncertainty due to a possible $M(X_{d,s})$-dependent
efficiency, the relative differences between the nominal selection
efficiencies and the ones obtained by reweighting the measured efficiencies by
the $X_{d,s}$ mass spectra in data are computed. The relative deviations of
0.5%, 1.1% and 1.2% for $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{+}_{s}K^{-}\pi^{+}\pi^{-}$, $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{+}_{s}\pi^{-}\pi^{+}\pi^{-}$ and $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow
D^{+}_{s}K^{-}\pi^{+}\pi^{-}$, respectively, are the assigned uncertainties.
The systematic uncertainty on the BDT efficiency is determined by fitting the
$\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{+}_{s}\pi^{-}\pi^{+}\pi^{-}$ mass distribution in data with and without the
BDT requirement. The efficiency is found to agree with simulation to better
than the 1% uncertainty assigned to this source. In total, the simulated
efficiencies have uncertainties of 1.6% and 1.9% in the two ratios of
branching fractions. The PID efficiency uncertainty is dominated by the usage
of the $D^{*+}$ calibration sample to determine the efficiencies of a given
PID requirement [34]. This uncertainty is assessed by comparing the PID
efficiencies obtained directly from simulated signal decays with the values
obtained using a simulated $D^{*+}$ calibration sample that is re-weighted to
match the kinematics of the signal decay particles. Using this technique, an
uncertainty of 2% each on the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{+}_{s}K^{-}\pi^{+}\pi^{-}$ and $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow
D^{+}_{s}K^{-}\pi^{+}\pi^{-}$ PID efficiencies is obtained, which is 100%
correlated, and a 1% uncertainty for $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{+}_{s}\pi^{-}\pi^{+}\pi^{-}$. The trigger is fully simulated, and given the
identical number of tracks and the well-modeled $p_{\rm T}$ spectra, the
associated uncertainty cancels to first order. Based on previous studies [16],
a 2% uncertainty is assigned.

The uncertainties in the signal yield determinations have contributions from
both the background and signal modeling. The signal shape uncertainty was
estimated by varying all the fixed signal shape parameters one at a time by
one standard deviation, and adding the changes in yield in quadrature (0.5%).
A double Gaussian signal shape model was also tried, and the difference was
negligible. For the combinatorial background, the shape was modified from a
single exponential to either the sum of two exponentials, or a linear
function. For $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{+}_{s}\pi^{-}\pi^{+}\pi^{-}$, the difference in yield was 0.4%. For $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{+}_{s}K^{-}\pi^{+}\pi^{-}$, the maximum change was 4%, and for $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow
D^{+}_{s}K^{-}\pi^{+}\pi^{-}$, the maximum shift was 1%. In the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{(s)}\rightarrow
D^{+}_{s}K^{-}\pi^{+}\pi^{-}$ mass fit, the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{(s)}\rightarrow
D_{s}^{*+}K^{-}\pi^{+}\pi^{-}$ contribution was modeled using the shape from
the $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{+}_{s}\pi^{-}\pi^{+}\pi^{-}$ mass fit. To estimate an uncertainty from this
assumption, the data were fitted with the shape obtained from $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{*+}_{s}K^{-}\pi^{+}\pi^{-}$ simulation. A deviation of 5.5% in the fitted
$\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow
D^{+}_{s}K^{-}\pi^{+}\pi^{-}$ yield is found, with almost no change in the
$\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{+}_{s}K^{-}\pi^{+}\pi^{-}$ yield. The larger sensitivity on the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ yield than the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ yield arises because these
background contributions have a rising edge in the vicinity of the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ mass peak, which is far enough
below the $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ mass peak to
have negligible impact. These yield uncertainties are added in quadrature to
obtain the values shown in Table 3. The uncertainties due to the finite
simulation sample sizes are 3.0%.

Table 3: Summary of systematic uncertainties (in %) on the measurements of the ratios of branching fractions. Source | ${{\cal{B}}(\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow D^{+}_{s}K^{-}\pi^{+}\pi^{-})\over{\cal{B}}(\kern 1.25995pt\overline{\kern-1.25995ptB}{}^{0}_{s}\rightarrow D^{+}_{s}\pi^{-}\pi^{+}\pi^{-})}$ | ${{\cal{B}}(\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow D^{+}_{s}K^{-}\pi^{+}\pi^{-})\over{\cal{B}}(\kern 1.25995pt\overline{\kern-1.25995ptB}{}^{0}_{s}\rightarrow D^{+}_{s}K^{-}\pi^{+}\pi^{-})}$
---|---|---
$f_{s}/f_{d}$ | - | 7.9
$M(X_{s,d})>3~{}{\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$ | 2.2 | 7.0
Efficiency | 1.6 | 1.9
PID | 2.2 | 0.0
Trigger | 2.0 | 2.0
Signal yields | 4.0 | 6.9
Simulated sample size | 3.0 | 3.0
Total | 6.4 | 13.4

The major source of systematic uncertainty on the branching fraction for
$\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D_{s1}(2536)^{+}\pi^{-},~{}D_{s1}^{+}\rightarrow D^{+}_{s}\pi^{-}\pi^{+}$, is
from the relative efficiency (5%), and on the fraction of events with
$M>3~{}{\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$ (10%). This 10% uncertainty
is conservatively estimated by assuming a flat distribution in $M(X_{d})$ up
to 3 ${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$, and then a linear decrease to
zero at the phase space limit of $\sim$3.5
${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$. Other systematic uncertainties
related to the fit model are negligible. Thus in total, a systematic
uncertainty of 11% is assigned to the ratio ${\cal{B}}(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D_{s1}(2536)^{+}\pi^{-},~{}D_{s1}^{+}\rightarrow
D^{+}_{s}\pi^{-}\pi^{+})/{\cal{B}}(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{+}_{s}\pi^{-}\pi^{+}\pi^{-})$.

##  9 Results and summary

This paper reports the first observation of the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{+}_{s}K^{-}\pi^{+}\pi^{-}$, $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow
D^{+}_{s}K^{-}\pi^{+}\pi^{-}$ and $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D_{s1}(2536)^{+}\pi^{-},~{}D_{s1}^{+}\rightarrow D^{+}_{s}\pi^{-}\pi^{+}$
decays. The ratios of branching fractions are measured to be

$\displaystyle{{\cal{B}}(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{+}_{s}K^{-}\pi^{+}\pi^{-})\over{\cal{B}}(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{+}_{s}\pi^{-}\pi^{+}\pi^{-})}$ $\displaystyle=(5.2\pm 0.5\pm 0.3)\times
10^{-2}$ $\displaystyle{{\cal{B}}(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow
D^{+}_{s}K^{-}\pi^{+}\pi^{-})\over{\cal{B}}(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{+}_{s}K^{-}\pi^{+}\pi^{-})}$ $\displaystyle=0.54\pm 0.07\pm 0.07,$

and

$\displaystyle{{\cal{B}}(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D_{s1}(2536)^{+}\pi^{-},~{}D_{s1}^{+}\rightarrow
D^{+}_{s}\pi^{-}\pi^{+})\over{\cal{B}}(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{+}_{s}\pi^{-}\pi^{+}\pi^{-})}=(4.0\pm 1.0\pm 0.4)\times 10^{-3},$

where the uncertainties are statistical and systematic, respectively. The
$\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{+}_{s}K^{-}\pi^{+}\pi^{-}$ branching fraction is consistent with
expectations from Cabibbo suppression. This decay is particularly interesting
because it can be used in a time-dependent analysis to measure the CKM phase
$\gamma$. Additional studies indicate that this decay mode, with selections
optimized for only $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{+}_{s}K^{-}\pi^{+}\pi^{-}$, can contribute about an additional 35% more
signal events relative to the signal yield in $B_{s}^{0}\rightarrow
D_{s}^{\mp}K^{\pm}$ alone.

The $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow
D^{+}_{s}K^{-}\pi^{+}\pi^{-}$ branching fraction is about 50% of that for
$\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{+}_{s}K^{-}\pi^{+}\pi^{-}$. Compared to the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow D^{+}_{s}K^{-}$ decay
that proceeds only via a $W$-exchange diagram, where ${\cal{B}}(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow
D^{+}_{s}K^{-})/{\cal{B}}(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow D^{+}_{s}K^{-})\sim
0.1$ [26], the ratio ${\cal{B}}(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow
D^{+}_{s}K^{-}\pi^{+}\pi^{-})/{\cal{B}}(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{+}_{s}K^{-}\pi^{+}\pi^{-})$ is about 5 times larger. A consistent
explanation of this larger $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow
D^{+}_{s}K^{-}\pi^{+}\pi^{-}$ branching fraction is that only about 1/5 of the
rate is from the $W$-exchange process (Fig. 1(d)) and about 4/5 comes from the
diagrams shown in Figs. 1(e-f). The observed $M(X_{s})$, $M(K^{-}\pi^{+})$ and
$M(\pi^{+}\pi^{-})$ distributions in Fig. 6 also support this explanation, as
evidenced by the qualitative agreement with the simulation.

## 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), CIEMAT, IFAE and UAB (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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