EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)

 CERN-PH-EP-2013-038 LHCb-PAPER-2012-055 May 4, 2013

Observation of the

suppressed ADS modes
$B^{\pm}\rightarrow[\pi^{\pm}K^{\mp}\pi^{+}\pi^{-}]_{D}K^{\pm}$ and
$B^{\pm}\rightarrow[\pi^{\pm}K^{\mp}\pi^{+}\pi^{-}]_{D}\pi^{\pm}$

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

An analysis of $B^{\pm}\rightarrow DK^{\pm}$ and $B^{\pm}\rightarrow
D\pi^{\pm}$ decays is presented where the $D$ meson is reconstructed in the
four-body final state $K^{\pm}\pi^{\mp}\pi^{+}\pi^{-}$. Using LHCb data
corresponding to an integrated luminosity of $1.0{\rm\,fb}^{-1}$, first
observations are made of the suppressed ADS modes
$B^{\pm}\rightarrow[\pi^{\pm}K^{\mp}\pi^{+}\pi^{-}]_{D}K^{\pm}$ and
$B^{\pm}\rightarrow[\pi^{\pm}K^{\mp}\pi^{+}\pi^{-}]_{D}\pi^{\pm}$ with a
significance of $5.1\sigma$ and greater than $10\sigma$, respectively.
Measurements of $C\\!P$ asymmetries and $C\\!P$-conserving ratios of partial
widths from this family of decays are also performed. The magnitude of the
ratio between the suppressed and favoured $B^{\pm}\rightarrow DK^{\pm}$
amplitudes is determined to be $r^{K}_{B}=0.097\pm{0.011}$.

Submitted to Phys. Lett. B

© 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. Andreassen59, R.B.
Appleby53, O. Aquines Gutierrez10, F. Archilli18, A. Artamonov 34, M.
Artuso56, E. Aslanides6, G. Auriemma24,m, S. Bachmann11, J.J. Back47, C.
Baesso57, 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, M.
Benayoun8, G. Bencivenni18, S. Benson49, J. Benton45, A. Berezhnoy31, R.
Bernet39, M.-O. Bettler46, M. van Beuzekom40, A. Bien11, S. Bifani12, T.
Bird53, A. Bizzeti17,h, P.M. Bjørnstad53, T. Blake37, F. Blanc38, J. Blouw11,
S. Blusk56, V. Bocci24, A. Bondar33, N. Bondar29, W. Bonivento15, S. Borghi53,
A. Borgia56, T.J.V. Bowcock51, E. Bowen39, C. Bozzi16, T. Brambach9, J. van
den Brand41, J. Bressieux38, D. Brett53, M. Britsch10, T. Britton56, 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, A. Carbone14,c, G. Carboni23,k, R. Cardinale19,i,
A. Cardini15, H. Carranza-Mejia49, L. Carson52, K. Carvalho Akiba2, G.
Casse51, M. Cattaneo37, Ch. Cauet9, M. Charles54, Ph. Charpentier37, P.
Chen3,38, N. Chiapolini39, M. Chrzaszcz 25, K. Ciba37, X. Cid Vidal36, 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, A. Cook45, M. Coombes45, S. Coquereau8, G. Corti37, B.
Couturier37, G.A. Cowan38, D. Craik47, S. Cunliffe52, R. Currie49, C.
D’Ambrosio37, P. David8, P.N.Y. David40, I. De Bonis4, K. De Bruyn40, S. De
Capua53, M. De Cian39, J.M. De Miranda1, M. De Oyanguren Campos35,o, L. De
Paula2, W. De Silva59, P. De Simone18, D. Decamp4, M. Deckenhoff9, L. Del
Buono8, D. Derkach14, O. Deschamps5, F. Dettori41, A. Di Canto11, 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, I. El
Rifai5, Ch. Elsasser39, D. Elsby44, A. Falabella14,e, C. Färber11, G.
Fardell49, C. Farinelli40, S. Farry12, V. Fave38, D. Ferguson49, V. Fernandez
Albor36, F. Ferreira Rodrigues1, M. Ferro-Luzzi37, S. Filippov32, C.
Fitzpatrick37, M. Fontana10, F. Fontanelli19,i, R. Forty37, O. Francisco2, M.
Frank37, C. Frei37, M. Frosini17,f, S. Furcas20, E. Furfaro23, A. Gallas
Torreira36, D. Galli14,c, M. Gandelman2, P. Gandini54, Y. Gao3, J. Garofoli56,
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. Göbel57, 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, O.
Grünberg58, B. Gui56, E. Gushchin32, Yu. Guz34, T. Gys37, C. Hadjivasiliou56,
G. Haefeli38, C. Haen37, S.C. Haines46, S. Hall52, T. Hampson45, S. Hansmann-
Menzemer11, N. Harnew54, S.T. Harnew45, J. Harrison53, T. Hartmann58, J. He7,
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, F. Jing3, M. John54, D. Johnson54, C.R. Jones46, 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.
Komarov38,31, 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. Lanfranchi18,37, C. Langenbruch37, T. Latham47,
C. Lazzeroni44, R. Le Gac6, J. van Leerdam40, J.-P. Lees4, R. Lefèvre5, A.
Leflat31,37, J. Lefrançois7, S. Leo22, O. Leroy6, B. Leverington11, Y. Li3, L.
Li Gioi5, M. Liles51, R. Lindner37, C. Linn11, B. Liu3, G. Liu37, J. von
Loeben20, S. Lohn37, 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, R. McNulty12,
A. Mcnab53, B. Meadows59,54, F. Meier9, M. Meissner11, M. Merk40, D.A.
Milanes8, M.-N. Minard4, J. Molina Rodriguez57, S. Monteil5, D. Moran53, P.
Morawski25, M.J. Morello22,s, R. Mountain56, 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,37, M.
Orlandea28, J.M. Otalora Goicochea2, P. Owen52, B.K. Pal56, 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. Phan56, E. Picatoste Olloqui35,
B. Pietrzyk4, T. Pilař47, D. Pinci24, S. Playfer49, M. Plo Casasus36, F.
Polci8, G. Polok25, A. Poluektov47,33, E. Polycarpo2, 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,
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. Shapoval42,37, P.
Shatalov30, Y. Shcheglov29, T. Shears51,37, L. Shekhtman33, O. Shevchenko42,
V. Shevchenko30, A. Shires52, R. Silva Coutinho47, T. Skwarnicki56, N.A.
Smith51, E. Smith54,48, M. Smith53, M.D. Sokoloff59, F.J.P. Soler50, F.
Soomro18,37, D. Souza45, B. Souza De Paula2, B. Spaan9, A. Sparkes49, P.
Spradlin50, F. Stagni37, S. Stahl11, O. Steinkamp39, S. Stoica28, S. Stone56,
B. Storaci39, M. Straticiuc28, U. Straumann39, V.K. Subbiah37, 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. Tobin39, 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. Voß58, H. Voss10, R. Waldi58, R. Wallace12,
S. Wandernoth11, J. Wang56, 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. Xing56, Z. Yang3, R. Young49, X. Yuan3, O.
Yushchenko34, M. Zangoli14, M. Zavertyaev10,a, F. Zhang3, L. Zhang56, 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, 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
56Syracuse University, Syracuse, NY, United States
57Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), Rio de
Janeiro, Brazil, associated to 2
58Institut für Physik, Universität Rostock, Rostock, Germany, associated to 11
59University of Cincinnati, Cincinnati, OH, United States, associated to 56

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

A measurement of the angle $\gamma$ (also denoted as $\phi_{3}$) of the CKM
Unitarity Triangle [1, *Kobayashi:1973fv] in processes involving tree-level
decays provides a Standard Model (SM) benchmark against which observables more
sensitive to new physics contributions can be compared. Currently such
comparisons are limited by the relatively large uncertainty ($\sim 12^{\circ}$
[3, *Bona:2005vz]) on the determination of $\gamma$ in tree-level decays [5,
6]. More precise measurements are therefore required.

A powerful strategy to measure $\gamma$ in tree-level processes is to study
$C\\!P$-violating observables in the decays $B^{\pm}\rightarrow Dh^{\pm}$,
where $D$ indicates a neutral charm meson which decays in a mode common to
both $D^{0}$ and $\kern 1.99997pt\overline{\kern-1.99997ptD}{}^{0}$ states,
and $h$, the bachelor hadron, is either a kaon or a pion. In the case of
$B^{-}\rightarrow DK^{-}$, interference occurs between the suppressed
$b\rightarrow u\bar{c}s$ and favoured $b\rightarrow c\bar{u}s$ decay paths,
and similarly for the charge conjugate decay. The magnitude of the
interference is governed by three parameters: the weak-phase difference,
$\gamma$, the $C\\!P$-conserving strong-phase difference, $\delta^{K}_{B}$,
and the ratio of the magnitudes of the two amplitudes, $r^{K}_{B}$. Similar
interference effects occur in the case when the bachelor hadron is a pion, but
additional Cabibbo suppression factors mean that the sensitivity to $\gamma$
is much diluted. Many possibilities exist for the $D$ decay mode, including
$C\\!P$ eigenstates [7, *Gronau:1991dp] and self-conjugate three-body decays
[9, *GGSZ2], which have both been exploited by LHCb in recent measurements
[11, 12]. Results of LHCb have also been presented making use of a similar
strategy with $B^{0}/\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ mesons
[13]. Another option, termed the ‘ADS’ method in reference to its originators
[14, *Atwood:2000ck], is to consider modes such as $D\rightarrow
K^{\pm}\pi^{\mp}$ and to focus on the suppressed final-state
$B^{\pm}\rightarrow[\pi^{\pm}K^{\mp}]_{D}K^{\pm}$, in which the favoured
$B^{\pm}$ decay is followed by a doubly Cabibbo-suppressed $D$ decay, or the
suppressed $B^{\pm}$ decay precedes a favoured $D$ decay. The amplitudes of
such combinations are of similar total magnitude and hence large interference
can occur, giving rise to significant $C\\!P$-violating effects. In contrast,
the interference in the favoured decay
$B^{\pm}\rightarrow[K^{\pm}\pi^{\mp}]_{D}K^{\pm}$ is low.

In this Letter, a search is performed for the previously unobserved ADS decays
$B^{\pm}\rightarrow[\pi^{\pm}K^{\mp}\pi^{+}\pi^{-}]_{D}h^{\pm}$. The $D$ decay
is treated inclusively, with no attempt to separate out the intermediate
resonances contributing to the four-body final state. LHCb has already
presented an ADS study using $B^{\pm}\rightarrow[\pi^{\pm}K^{\mp}]_{D}h^{\pm}$
decays, and many features of the current analysis are similar to those of the
earlier paper [11]. In this study a total of seven observables is measured:
the ratio of partial widths involving the favoured modes

$R_{K/\pi}^{K3\pi}\equiv\frac{\Gamma(B^{-}\rightarrow[K^{-}\pi^{+}\pi^{-}\pi^{+}]_{D}K^{-})+\Gamma(B^{+}\rightarrow[K^{+}\pi^{-}\pi^{+}\pi^{-}]_{D}K^{+})}{\Gamma(B^{-}\rightarrow[K^{-}\pi^{+}\pi^{-}\pi^{+}]_{D}\pi^{-})+\Gamma(B^{+}\rightarrow[K^{+}\pi^{-}\pi^{+}\pi^{-}]_{D}\pi^{+})},$
(1)

two $C\\!P$ asymmetries, again involving the favoured modes

$A_{h}^{K3\pi}\equiv\frac{\Gamma(B^{-}\rightarrow[K^{-}\pi^{+}\pi^{+}\pi^{-}]_{D}h^{-})-\Gamma(B^{+}\rightarrow[K^{+}\pi^{-}\pi^{+}\pi^{-}]_{D}h^{+})}{\Gamma(B^{-}\rightarrow[K^{-}\pi^{+}\pi^{+}\pi^{-}]_{D}h^{-})+\Gamma(B^{+}\rightarrow[K^{+}\pi^{-}\pi^{+}\pi^{-}]_{D}h^{+})},$
(2)

and four charge-separated partial widths of the suppressed ADS mode relative
to the favoured mode

$R_{h}^{K3\pi,\pm}\equiv\frac{\Gamma(B^{\pm}\rightarrow[\pi^{\pm}K^{\mp}\pi^{+}\pi^{-}]_{D}h^{\pm})}{\Gamma(B^{\pm}\rightarrow[K^{\pm}\pi^{\mp}\pi^{+}\pi^{-}]_{D}h^{\pm})}.$
(3)

The observables $R_{h}^{K3\pi,\pm}$ carry the highest sensitivity to $\gamma$,
$r_{B}^{h}$ and $\delta_{B}^{h}$. They are related by the expression [14,
*Atwood:2000ck, 16]

$R_{h}^{K3\pi,\pm}={(r^{h}_{B})^{2}+(r_{D}^{K3\pi})^{2}+2r^{h}_{B}r^{K3\pi}_{D}\kappa_{D}^{K3\pi}\cos(\delta_{B}^{h}+\delta_{D}^{K3\pi}\pm\gamma}).$
(4)

Here $r_{D}^{K3\pi}$ is the ratio of the magnitudes of the doubly Cabibbo-
suppressed and Cabibbo-favoured $D$ decay amplitudes, and $\delta_{D}^{K3\pi}$
is the strong-phase difference between the amplitudes, averaged over the
final-state phase space. The coherence factor $\kappa_{D}^{K3\pi}$ accounts
for possible dilution effects in the interference arising from the
contribution of several intermediate resonances in the $D$ decay [16].
Information on these $D$ decay parameters is available from external sources.
Branching ratio measurements indicate that $r^{K3\pi}_{D}\sim 0.06$ [17].
Studies of quantum-correlated $D\kern 1.99997pt\overline{\kern-1.99997ptD}{}$
pairs, performed at CLEO-c, yield
$\delta_{D}^{K3\pi}=(114\,^{+26}_{-23})^{\circ}$ and
$\kappa_{D}^{K3\pi}=0.33\,^{+0.20}_{-0.23}$ [18].111The phase
$\delta_{D}^{K3\pi}$ is given in the convention where
$C\\!P|D^{0}\rangle=|\kern 1.99997pt\overline{\kern-1.99997ptD}{}^{0}\rangle$.
The relatively low value of the coherence factor limits the sensitivity of
$R_{h}^{K3\pi,\pm}$ to $\gamma$ and $\delta_{B}^{h}$, but does not hinder this
observable in providing information on $r_{B}^{h}$. Improved knowledge of
$r_{B}^{h}$ is valuable in providing a constraint which other
$B^{\pm}\rightarrow Dh^{\pm}$ analyses can benefit from.

##  2 The LHCb detector and the analysis sample

This analysis uses data, corresponding to an integrated luminosity of 1.0
$\mbox{\,fb}^{-1}$, collected by LHCb in 2011 at
$\sqrt{s}=7~{}\mathrm{\,Te\kern-1.00006ptV}$. The LHCb experiment [19] takes
advantage of the high $b\bar{b}$ and $c\bar{c}$ production cross sections at
the Large Hadron Collider to record large samples of heavy-hadron decays. It
instruments the pseudorapidity range $2<\eta<5$ of the proton-proton ($pp$)
collisions with a dipole magnet and a tracking system that achieves a momentum
resolution of $0.4-0.6\%$ in the range $5-100$
${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$. The dipole magnet can be operated in
either polarity and this feature is used to reduce systematic effects due to
detector asymmetries. In 2011, 58% of the data were taken with one polarity,
42% with the other. The $pp$ collisions take place inside a silicon microstrip
vertex detector that provides clear separation of secondary $B^{\pm}$ vertices
from the primary collision vertex (PV) as well as discrimination for tertiary
$D$ vertices. Two ring-imaging Cherenkov (RICH) detectors [20] with three
radiators (aerogel, ${\rm C_{4}F_{10}}$ and ${\rm CF_{4}}$) provide dedicated
particle identification (PID), which is critical for the separation of
$B^{\pm}\\!\rightarrow\\!DK^{\pm}$ and $B^{\pm}\\!\rightarrow\\!D\pi^{\pm}$
decays.

A two-stage trigger is employed. First, a hardware-based decision is taken at
a rate of up to 40 MHz. It accepts high transverse energy clusters in either
an electromagnetic or hadron calorimeter, or a muon of high momentum
transverse to the beam line ($p_{\rm T}$). For this analysis, it is required
that either one of the five tracks forming the $B^{\pm}$ candidate points at a
cluster in the hadron calorimeter, or that the hardware-trigger decision was
taken independently of any of these tracks. A subsequent trigger level,
implemented entirely in software, receives events at a rate of 1 MHz and
retains $\sim 0.3\%$ of them. At least one track should have $\mbox{$p_{\rm
T}$}>1.7$ ${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ and impact parameter (IP)
$\chi^{2}$ with respect to the PV greater than 16. The IP $\chi^{2}$ is
defined as the difference between the $\chi^{2}$ of the PV reconstructed with
and without the considered track. In order to maximise efficiency at an
acceptable trigger rate, a displaced vertex is selected with a decision tree
algorithm that uses flight distance as well as fit quality, $p_{\rm T}$ and
information on the IP with respect to the PV of the tracks. More information
can be found in Ref. [21]. Full event reconstruction occurs offline, and a
loose selection is run to reduce the size of the sample prior to final
analysis. This selection consists of a decision tree algorithm similar to that
used in the trigger, but in this case the entire decay chain is fully
reconstructed and the selection benefits from the improved quality of the
offline reconstruction.

Approximately one million simulated signal events are used in the analysis as
well as a sample of $\sim 10^{8}$ generic $B_{q}\rightarrow DX$ decays, where
$q\in\\{u,d,s\\}$. These samples are generated using Pythia 6.4 [22]
configured with parameters detailed in Ref. [23]. The EvtGen package [24] is
used to generate hadronic decays, in which final state radiation is generated
using the Photos package [25]. The interaction of the generated particles with
the LHCb detector is simulated using the Geant4 toolkit [26,
*Agostinelli:2002hh] as described in Ref. [28].

##  3 Candidate selection and background rejection

The reconstruction considers all $B^{\pm}\rightarrow Dh^{\pm}$ channels of
interest. The reconstructed $D$ candidate mass is required to be within $\pm
25{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ ($\approx 3.5\sigma$) of its
nominal value [17]. The $D$ daughter tracks are required to have
$\mbox{$p_{\rm T}$}>0.25$${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$, while the
bachelor track is required to satisfy $0.5<\mbox{$p_{\rm
T}$}<10$${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ and
$5<p<100$${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$. The tighter requirements on
the bachelor track ensure that it resides within the kinematic coverage of the
PID calibration samples acquired through the decay mode $D^{*+}\rightarrow
D^{0}\pi^{+},D^{0}\rightarrow K^{-}\pi^{+}$. Details of the PID calibration
procedure are given in Sect. 4. Furthermore, a kinematic fit is performed to
each decay chain [29] constraining both the $B^{\pm}$ and $D$ vertices to
points in 3D space, while simultaneously constraining the $D$ candidate to its
nominal mass. This fit results in a $B^{\pm}$ mass resolution of
15${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$, a 10% improvement with respect
to the value prior to the fit. Candidates are retained that have an invariant
mass in the interval $5120-5750$${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$.

A boosted decision tree (BDT) discriminator [30], implementing the
_GradientBoost_ algorithm [31], is employed to achieve further background
suppression. The BDT is trained using the simulated $B^{\pm}\rightarrow
Dh^{\pm}$ events together with a pure sample of combinatoric background
candidates taken from a subset of the data in the invariant mass range
$5500-5800$${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$. The BDT considers a
variety of properties associated with each signal candidate. These properties
can be divided into two categories; (i) quantities common to both the tracks
and to the $D$ and $B^{\pm}$ candidates, (ii) quantities associated with only
the $D$ and $B^{\pm}$ candidates. Specifically, the properties considered in
each case are as follows:

  1. (i)

$p$, $p_{\rm T}$ and IP $\chi^{2}$;

  2. (ii)

decay time, flight distance from the PV, vertex quality, and the angle between
the particle’s momentum vector and a line connecting the PV to the particle’s
decay vertex.

In addition, the BDT also considers information from the rest of the event
through an isolation variable that represents the imbalance of $p_{\rm T}$
around the $B^{\pm}$ candidate. The variable is defined as

$A_{\mbox{$p_{\rm T}$}}=\frac{\mbox{$p_{\rm T}$}(B^{\pm})-\sum\mbox{$p_{\rm
T}$}}{\mbox{$p_{\rm T}$}(B^{\pm})+\sum\mbox{$p_{\rm T}$}},$ (5)

where $\sum\mbox{$p_{\rm T}$}$ corresponds to the sum of $p_{\rm T}$ over all
tracks identified, within a cone of half-angle 1.5 in pseudorapidity and 1.5
rad in azimuthal angle, that are not associated with the signal $B^{\pm}$
candidate. Since no PID information is used as input during the training, the
BDT has very similar performance for both $B^{\pm}\rightarrow DK^{\pm}$ and
$B^{\pm}\rightarrow D\pi^{\pm}$ decay modes, with small differences arising
from the variation in kinematics between the two.

The optimal cut value on the BDT response is chosen by considering the
combinatorial background level ($b$) in the invariant mass distribution of the
favoured $B^{\pm}\rightarrow D\pi^{\pm}$ final state. The large signal peak in
this sample is scaled by the predicted branching fraction of the suppressed
mode for the case when the interference amongst the intermediate resonances of
the $D$ decay is maximally destructive. This provides a conservative estimate
of the suppressed-sign signal yield ($s$). It is then possible to construct
the quantity $s/\sqrt{s+b}$ to serve as an optimisation metric. Assessment of
this metric finds an optimal working point where a signal efficiency of $\sim
85\%$ is expected while rejecting $>99\%$ of the combinatorial background.
This same working point is used in selecting both suppressed and favoured
final states.

PID information is quantified as differences between the logarithm of
likelihoods, $\ln\mathcal{L}_{h}$, under five mass hypotheses,
$h\in\\{\pi,K,p,e,\mu\\}$ (DLL). The daughter kaon from the $D$ meson decay is
required to satisfy ${\rm
DLL}_{K\pi}\equiv\ln\mathcal{L}_{K}-\ln\mathcal{L}_{\pi}>2$, while the
daughter pions must have ${\rm DLL}_{K\pi}<-2$. A sample enriched in
$B^{\pm}\rightarrow DK^{\pm}$ decays is selected by requiring ${\rm
DLL}_{K\pi}>4$ for the bachelor hadron. Candidates failing this cut are
retained in a separate sample, which is predominantly composed of
$B^{\pm}\rightarrow D\pi^{\pm}$ decays.

Backgrounds from genuine $B^{\pm}$ decays that do not involve a true $D$ meson
are suppressed by requiring the flight distance significance of the $D$
candidate from the $B^{\pm}$ vertex be greater than $2$. The branching ratios
of five-body charmless decays are currently unmeasured, and so the residual
contamination from this source is estimated by assuming that the proportion of
these decays passing the
$B^{\pm}\rightarrow[\pi^{\pm}K^{\mp}\pi^{+}\pi^{-}]_{D}h^{\pm}$ selection is
the same as the proportion of three-body charmless decays passing the
$B^{\pm}\rightarrow[\pi^{\pm}K^{\mp}]_{D}h^{\pm}$ selection reported in Ref.
[11]. In the case of $B^{\pm}\rightarrow\pi^{+}\pi^{-}\pi^{+}\pi^{-}K^{\pm}$
this assumption can be validated by removing the flight significance cut and
inspecting the sideband above the $D$ mass after adjusting the selection to
isolate $B^{\pm}\rightarrow[\pi^{+}\pi^{-}\pi^{+}\pi^{-}]_{D}K^{\pm}$ decays.
A $B^{\pm}\rightarrow\pi^{+}\pi^{-}\pi^{+}\pi^{-}K^{\pm}$ signal is observed
with a magnitude compatible with that found when scaling the results of the
analogous exercise performed with a
$B^{\pm}\rightarrow[\pi^{+}\pi^{-}]_{D}K^{\pm}$ selection, and is eliminated
when the flight-significance cut is reinstated. Following these studies the
residual contamination from five-body charmless decays is determined to be
$2.2\pm 1.1$ candidates for the suppressed $B^{\pm}\rightarrow DK^{\pm}$
selection, and negligible for all other samples.

Contamination involving misidentified charmonium decays is eliminated by
considering the possible neutral combinations of the bachelor track and any
one of the $D$ daughter tracks under the hypothesis that both tracks are
muons. For those combinations where both tracks satisfy a loose muon PID
requirement, the parent $B^{\pm}$ candidate is vetoed if the invariant mass of
this combination is within $\pm 22$${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$
of either the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ or $\psi{(2S)}$
mass [17].

The suppressed signal sample suffers a potentially large cross-feed from
favoured signal decays in which a $K^{\pm}$ and $\pi^{\mp}$ from the $D$ decay
are misidentified as $\pi^{\pm}$ and $K^{\mp}$, respectively. This
contamination is reduced by vetoing any suppressed candidate whose
reconstructed $D$ mass, under the exchange of mass hypotheses between the
daughter kaon and either of the two same-sign daughter pions, lies within $\pm
15$${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ of the nominal $D$ mass. For the
measurement of $R_{h}^{K3\pi,\pm}$ this veto is also applied to the favoured
mode. Study of the cross-feed contamination in the mass sidebands of the $D$
candidates allows the estimate of the residual contamination in the signal
region to be checked. The residual cross-feed after all selection requirements
is estimated to be $(7.1\pm 3.1)\times 10^{-5}$.

The mass window for the $D$ candidates is sufficiently tight to eliminate
background arising from single-track misidentifications in the four-body
decays $D\rightarrow K^{+}K^{+}\pi^{+}\pi^{-}$ and
$D\rightarrow\pi^{+}\pi^{-}\pi^{+}\pi^{-}$. The good performance of the RICH
system ensures that the residual background from $D\rightarrow
K^{+}K^{-}\pi^{+}\pi^{-}$ decays in which three tracks are misidentified is
negligible. The contamination in the suppressed
$B^{\pm}\rightarrow[\pi^{\pm}K^{\mp}\pi^{+}\pi^{-}]_{D}K^{\pm}$ sample from
$B^{\pm}\rightarrow[K^{\mp}K^{0}_{\rm\scriptscriptstyle S}\pi^{-}]_{D}K^{\pm}$
with $K^{0}_{\rm\scriptscriptstyle S}\rightarrow\pi^{+}\pi^{-}$ is estimated
from simulation to occur at the rate of $(6.1\pm 1.7)\times 10^{-5}$.

Only one candidate per event is retained for analysis. In the $0.8\%$ of
events that contain more than one candidate a choice is made by selecting the
candidate with the best-quality $B^{\pm}$ vertex.

Using simulation it is found that the selection leads to an acceptance across
the four-body phase space of the $D$ decay that is uniform to a good
approximation. This property is important as it means that the values of the
coherence factor and strong-phase difference measured in Ref. [18], which are
integrated over all phase space, can be applied when interpreting the results
of the current analysis. It is verified using simulation that the small non-
uniformities that exist in the acceptance induce negligible bias in the
effective value of these parameters.

##  4 Signal yields and systematic uncertainties

The observables of interest are determined with a binned maximum-likelihood
fit to the invariant mass distributions of the selected $B^{\pm}$ candidates.
Distinguishing between $B^{+}$ and $B^{-}$ candidates, favoured and suppressed
decay topologies, and those that pass or fail the bachelor PID requirement
imposed to select $B^{\pm}\rightarrow DK^{\pm}$ decays, yields eight disjoint
subsamples, which are fitted simultaneously. The total probability density
function (PDF) used in the fit is built from four main sources representing
the various categories of candidates in each subsample.

  1. 1.

$B^{\pm}\rightarrow D\pi^{\pm}$
In the subsamples failing the bachelor PID cut, a modified Gaussian function,

$f(m)\propto\exp\left(\frac{-(m-\mu)^{2}}{2\sigma^{2}+(m-\mu)^{2}\alpha_{L,R}}\right)$
(6)

describes the asymmetric function of peak of value $\mu$ and width $\sigma$
where $\alpha_{L}(m<\mu)$ and $\alpha_{R}(m>\mu)$ parameterise the tails. True
$B^{\pm}\rightarrow D\pi^{\pm}$ candidates that pass the PID cut are
reconstructed as $B^{\pm}\rightarrow DK^{\pm}$. As these candidates have an
incorrect mass assignment they form a displaced mass peak with a tail that
extends to higher invariant mass. These candidates are modelled by the sum of
two Gaussian PDFs, also altered to include tail components as in Eq. 6. All
parameters are allowed to vary except the lower-mass tail which is fixed to
the value found in simulation to ensure fit stability, and later considered
amongst the systematic uncertainties. These shapes are considered identical
for $B^{+}$ and $B^{-}$ decays.

  2. 2.

$B^{\pm}\rightarrow DK^{\pm}$
In the subsamples that pass the PID cut on the bachelor, the same modified
Gaussian function as quoted in Eq. 6 is used. The peak value and the two tail
parameters are identical to those of the higher $B^{\pm}\rightarrow
D\pi^{\pm}$ peak. The width is $0.95\pm 0.02$ times the $B^{\pm}\rightarrow
D\pi^{\pm}$ width, as determined by a separate study of the favoured mode.
Candidates failing the PID cut are described by a fixed shape that is obtained
from simulation and later varied to assess the systematic uncertainty.

  3. 3.

Partially reconstructed ${b}$-hadron decays
Partially reconstructed decays populate the invariant mass region below the
$B^{\pm}$ mass. Such candidates may enter the signal region, especially where
Cabibbo-favoured $B^{\pm}\rightarrow XD\pi^{\pm}$ modes are misidentified as
$B^{\pm}\rightarrow DK^{\pm}$. The large simulated sample of inclusive
$B_{q}\rightarrow DX$ decays is used to model this background. After applying
the selection, two non-parametric PDFs [32] are defined (for the $D\pi^{\pm}$
and $DK^{\pm}$ selections) and used in the signal fit for both the favoured
and suppressed mode subsamples.

In addition, partially reconstructed $B^{0}_{s}\rightarrow DK^{-}\pi^{+}$ and
$\mathchar
28931\relax^{0}_{b}\rightarrow[pK^{-}\pi^{+}\pi^{-}\pi^{+}]_{\mathchar
28931\relax^{+}_{c}}h^{-}$ decays and their charge-conjugated modes are
considered as background sources specific to the suppressed
$B^{\pm}\rightarrow DK^{\pm}$ and favoured mode subsamples, respectively. PDFs
for both these sources of background are determined from simulation and
smeared to match the resolution observed in data.

The yield of these background components in each subsample varies
independently in the fit, making no assumption of $C\\!P$ symmetry.

  4. 4.

Combinatoric background
A linear approximation is adequate to describe the distribution across the
invariant mass spectrum considered. A common shape parameter is used in all
subsamples, though yields vary independently.

The proportion of $B^{\pm}\rightarrow Dh^{\pm}$ passing or failing the PID
requirement is determined from an analysis of approximately 20 million
$D^{*\pm}$ decays reconstructed as $D^{*\pm}\rightarrow
D\pi^{\pm},\,D\rightarrow K^{\mp}\pi^{\pm}$. The reconstruction is performed
using only kinematic variables, and provides a high purity calibration sample
of $K$ and $\pi$ tracks which is unbiased for studies exploiting the RICH and
is therefore made use of to measure the PID efficiency as a function of track
momentum, pseudorapidity and number of tracks in the detector. Through
reweighting the calibration spectra in these variables to match that of the
candidates in the $B^{\pm}\rightarrow D\pi^{\pm}$ peak, the effective PID
efficiency of the signal is determined. This data-driven approach finds a
retention rate on the bachelor track of 86.1% and 3.7% for kaons and pions,
respectively. An absolute $1.0\%$ systematic uncertainty on the kaon
efficiency is estimated from simulation. The $B^{\pm}\rightarrow D\pi^{\pm}$
fit to data becomes significantly incorrect when the PID efficiency is varied
outside the absolute range of $\pm 0.2\%$, and so this value is taken as the
systematic uncertainty for pions.

Detection and production asymmetries are accounted for using the same
procedure followed in Ref. [11], based on the measurement of the observed raw
asymmetry of $B^{\pm}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{\pm}$ decays in the LHCb detector [33]. A detection asymmetry of
$(-0.5\pm 0.7)$% is assigned for each unit of strangeness in the final state
to account for the different interaction lengths of $K^{-}$ and $K^{+}$
mesons. The equivalent asymmetry for pions is expected to be much smaller and
($0.0\pm 0.7$)% is assigned. This uncertainty also accounts for the residual
physical asymmetry between the left and right sides of the detector after
summing both magnet-polarity data sets. Simulation of $b$-hadron production in
$pp$ collisions suggests a small excess of $B^{+}$ over $B^{-}$ mesons. A
production asymmetry of $(-0.8\pm 0.7)$% is assumed in the fit such that the
combination of these estimates aligns with the observed raw asymmetry of
$B^{\pm}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{\pm}$ decays
[33].

The signal yields for the favoured and suppressed $B^{\pm}\rightarrow
Dh^{\pm}$ decays, after summing the events that pass and fail the bachelor PID
cut, are shown in Table 1. Their corresponding invariant mass spectra,
separated by the charge of the $B$ candidate, are shown in Figs. 1 and 2.
Plots of the combined $B^{+}$ and $B^{-}$ suppressed-mode mass spectra are
shown in Fig. 3.

Table 1: Favoured ($[K\pi\pi\pi]_{D}h$) and suppressed ($[\pi K\pi\pi]_{D}h$) signal yields together with their corresponding statistical uncertainties.  Mode | $B^{-}$ | $B^{+}$
---|---|---
$[K\pi\pi\pi]_{D}\pi$ | $20,791\pm 232$ | $21,054\pm 235$
$[K\pi\pi\pi]_{D}K$ | $\phantom{2}1,567\pm\phantom{0}57$ | $\phantom{2}1,660\pm\phantom{0}60$
$[\pi K\pi\pi]_{D}\pi$ | $\phantom{20,7}87\pm\phantom{0}11$ | $\phantom{21,0}68\pm\phantom{0}10$
$[\pi K\pi\pi]_{D}K$ | $\phantom{20,7}11\pm\phantom{00}5$ | $\phantom{21,0}29\pm\phantom{00}7$
Figure 1: Invariant mass distributions of selected
$B^{\pm}\rightarrow[K^{\pm}\pi^{\mp}\pi^{+}\pi^{-}]_{D}h^{\pm}$ candidates,
separated by charge. The left plots are $B^{-}$ candidates, $B^{+}$ are on the
right. In the top plots, the bachelor track passes the PID cut and the
$B^{\pm}$ candidates are reconstructed assigning this track the kaon mass. The
remaining candidates are placed in the sample displayed on the bottom row and
are reconstructed with a pion mass hypothesis. The dark (red) and light
(green) curves represent the fitted $B^{\pm}\rightarrow DK^{\pm}$ and
$B^{\pm}\rightarrow D\pi^{\pm}$ components, respectively. The shaded
contribution indicates partially reconstructed decays and the total PDF
includes the combinatorial component.  Figure 2: Invariant mass distributions
of selected $B^{\pm}\rightarrow[\pi^{\pm}K^{\mp}\pi^{+}\pi^{-}]_{D}h^{\pm}$
decays, separated by charge. See the caption of Fig. 1 for a full description.
The dashed line here represents the partially reconstructed, but Cabibbo
favoured, $B^{0}_{s}\rightarrow DK^{-}\pi^{+}$, and charge-conjugated, decays
where the pion is not reconstructed. The favoured mode cross-feed is included
in the fit, but is too small to be seen.  Figure 3: Charge-integrated
invariant mass distributions of those candidates shown in Fig. 2 passing the
$B^{\pm}\rightarrow[\pi^{\pm}K^{\mp}\pi^{+}\pi^{-}]_{D}h^{\pm}$ selection.

The $R^{K3\pi,\pm}_{h}$ observables are related to the ratio of event yields
by the relative efficiency, independent of PID effects, with which
$B^{\pm}\rightarrow DK^{\pm}$ and $B^{\pm}\rightarrow D\pi^{\pm}$ decays are
reconstructed. This ratio is found to be 1.029 from a simulation study. A 2.4%
systematic uncertainty, based on the finite size of the simulated sample,
accounts for the imperfect modelling of the relative pion and kaon absorption
in the tracking material.

The fit is constructed such that the observables of interest are free
parameters. To estimate the systematic uncertainties arising from the
imperfect knowledge of several of the external parameters discussed above, the
fit is performed many times varying each input by its assigned error. The
resulting spread (RMS) in the value of each observable is taken as the
systematic uncertainty on that quantity and is summarised in Table 2.
Correlations between the uncertainties are considered negligible, so the total
systematic uncertainty is the sum in quadrature of the individual components.

Table 2: Systematic uncertainties on the observables. Bachelor PID refers to the fixed efficiency for the bachelor track ${\rm DLL}_{K\pi}$ requirement determined using the $D^{*\pm}$ calibration sample. PDFs refers to the variations of the fixed shapes in the mass fit. Simulation refers to the use of simulation to estimate relative efficiencies of the signal modes, and also includes the contribution from the uncertainty in the residual background from charmless $B$ decays. $A_{\rm instr.}$ quantifies the uncertainty on the production, interaction and detection asymmetries.  $[\times 10^{-3}]$ | $R_{K/\pi}^{K3\pi}$ | $A_{\pi}^{K3\pi}$ | $A_{K}^{K3\pi}$ | $R_{K}^{K3\pi,-}$ | $R_{K}^{K3\pi,+}$ | $R_{\pi}^{K3\pi,-}$ | $R_{\pi}^{K3\pi,+}$
---|---|---|---|---|---|---|---
Bachelor PID | 1.7 | 0.2 | 0.6 | 0.4 | 0.4 | 0.02 | 0.04
PDFs | 1.2 | 1.3 | 4.4 | 0.7 | 0.9 | 0.09 | 0.08
Simulation | 1.5 | 0.1 | 0.3 | 0.1 | 0.2 | 0.01 | 0.02
$A_{\rm instr.}$ | 0.0 | 9.9 | 17.1 | 0.1 | 0.1 | 0.06 | 0.06
Total | 2.6 | 10.0 | 17.7 | 0.8 | 1.0 | 0.11 | 0.11

##  5 Results and interpretation

The results of the fit with their statistical and systematic uncertainties are

$\begin{array}[]{l c<EMAIL_ADDRESS>l}R_{K/\pi}^{K3\pi}&=&0&0771\phantom{0}\hskip
5.0pt\pm\hskip 5.0pt0.0017&\pm\hskip 5.0pt0.0026\\\\[8.0pt]
A_{K}^{K3\pi}&=&-0&029\phantom{00}\hskip 5.0pt\pm\hskip
5.0pt0.020\phantom{0}&\pm\hskip 5.0pt0.018\\\\[8.0pt]
A_{\pi}^{K3\pi}&=&-0&006\phantom{00}\hskip 5.0pt\pm\hskip
5.0pt0.005\phantom{0}&\pm\hskip 5.0pt0.010\\\\[8.0pt]
R_{K}^{K3\pi,-}&=&0&0072\phantom{0}^{\;\;\;\;+\;\;\;\;0.0036}_{\;\;\;\;-\;\;\;\;0.0032}&\pm\hskip
5.0pt0.0008\\\\[8.0pt]
R_{K}^{K3\pi,+}&=&0&0175\phantom{0}^{\;\;\;\;+\;\;\;\;0.0043}_{\;\;\;\;-\;\;\;\;0.0039}&\pm\hskip
5.0pt0.0010\\\\[8.0pt]
R_{\pi}^{K3\pi,-}&=&0&00417^{\;\;\;\;+\;\;\;\;0.00054}_{\;\;\;\;-\;\;\;\;0.00050}&\pm\hskip
5.0pt0.00011\\\\[8.0pt]
R_{\pi}^{K3\pi,+}&=&0&00321^{\;\;\;\;+\;\;\;\;0.00048}_{\;\;\;\;-\;\;\;\;0.00045}&\pm\hskip
5.0pt0.00011\ .\\\\[5.0pt] \end{array}$

From these measurements, the quantities $R^{K3\pi}_{{\rm ADS}(h)}$ and
$A^{K3\pi}_{{\rm ADS}(h)}$ can be deduced. These are, respectively, the ratio
of the suppressed to the favoured partial widths for the decays
$B^{\pm}\rightarrow Dh^{\pm}$, averaged over the two charges, and the $C\\!P$
asymmetry of the suppressed decay mode

$\displaystyle R^{K3\pi}_{{\rm ADS}(K)}$ $\displaystyle=$
$\displaystyle\frac{R_{K}^{K3\pi,-}+R_{K}^{K3\pi,+}}{2}=\phantom{-}0.0124\hskip
5.0pt\pm\hskip 5.0pt0.0027$ $\displaystyle A^{K3\pi}_{{\rm ADS}(K)}$
$\displaystyle=$
$\displaystyle\frac{R_{K}^{K3\pi,-}-R_{K}^{K3\pi,+}}{R_{K}^{K3\pi,-}+R_{K}^{K3\pi,+}}=-0.42\phantom{00}\hskip
5.0pt\pm\hskip 5.0pt0.22$ $\displaystyle R^{K3\pi}_{{\rm ADS}(\pi)}$
$\displaystyle=$
$\displaystyle\frac{R_{\pi}^{K3\pi,-}+R_{\pi}^{K3\pi,+}}{2}=\phantom{-}0.0037\hskip
5.0pt\pm\hskip 5.0pt0.0004$ $\displaystyle A^{K3\pi}_{{\rm ADS}(\pi)}$
$\displaystyle=$
$\displaystyle\frac{R_{\pi}^{K3\pi,-}-R_{\pi}^{K3\pi,+}}{R_{\pi}^{K3\pi,-}+R_{\pi}^{K3\pi,+}}=\phantom{-}0.13\phantom{00}\hskip
5.0pt\pm\hskip 5.0pt0.10\ .$

The displayed uncertainty is the combination of statistical and systematic
contributions. Correlations between systematic uncertainties are taken into
account in the combination. It can be seen that the observable
$A^{K3\pi}_{{\rm ADS}(K)}$, which is expected to manifest significant $C\\!P$
violation, differs from the $C\\!P$-conserving hypothesis by around $2\sigma$.

A likelihood ratio test is employed to assess the significance of the
suppressed ADS signal yields. This has been performed calculating the quantity
$\sqrt{-2{\rm ln}\frac{\mathcal{L}_{\rm b}}{\mathcal{L}_{\rm s+b}}}$, where
$\mathcal{L}_{\rm s+b}$ and $\mathcal{L}_{\rm b}$ are the maximum values of
the likelihoods in the case of a signal-plus-background and background-only
hypothesis, respectively. Significances of $5.7\sigma$ and greater than
$10\sigma$ are determined for the modes
$B^{\pm}\rightarrow[\pi^{\pm}K^{\pm}\pi^{+}\pi^{-}]_{D}K^{\pm}$ and
$B^{\pm}\rightarrow[\pi^{\pm}K^{\mp}\pi^{+}\pi^{-}]_{D}\pi^{\pm}$,
respectively. The former significance is found to reduce to $5.1\sigma$ when
the systematic uncertainties are included.

The measured observables are used to infer a confidence interval for the value
of the suppressed-to-favoured $B^{\pm}\rightarrow DK^{\pm}$ amplitude ratio,
$r^{K}_{B}$. The most probable value of $r^{K}_{B}$ is identified as that
which minimises the $\chi^{2}$ calculated from the measured observables and
their predictions for the given value of $r^{K}_{B}$. The prediction for
$R^{K3\pi,\pm}_{K}$ is given by Eq. 4, and similar relations exist for the
other observables. Amongst the other parameters that determine the predicted
values, $r_{B}^{h}$, $\delta_{B}^{h}$ and $\gamma$ vary freely, but all the
parameters of the $D$ decay, notably the coherence factor and strong-phase
difference, are constrained by the results in Ref. [18]. Subsequently, the
evolution of the minimum $\chi^{2}$ is inspected across the range
$(0.0<r^{K}_{B}<0.2)$ and the difference $\Delta\chi^{2}$ with respect to the
global minimum is calculated. The probabilistic interpretation of the
$\Delta\chi^{2}$ at each value of $r^{K}_{B}$ is evaluated by generating and
fitting a large number ($10^{7}$) of pseudo-datasets around the local minimum.
The variation of the pseudo-datasets is derived from the covariance matrix of
the principal result. At a given fixed point in the $r_{B}^{K}$ range, $a$,
with a value of $\Delta\chi^{2}_{a}$ above the global minimum, the probability
of obtaining the observed result is defined as the number of pseudo-
experiments with $\Delta\chi^{2}\geq\Delta\chi^{2}_{a}$. By this frequentist
technique it is found that the result for $r^{K}_{B}$ has a non-Gaussian
uncertainty, so the “$1\sigma$” and “$2\sigma$” intervals, respectively, are
given as

$r^{K}_{B}=0.097\pm 0.011\ [68.3\%{\rm\ CL}]\,\,\,{\rm
and}\,\,\,^{+0.027}_{-0.029}\ [95.5\%{\rm\ CL}]\ .$

The measurements do not allow significant constraints to be set on the other
underlying physics parameters.

##  6 Conclusions

A search has been performed for the ADS suppressed modes
$B^{\pm}\rightarrow[\pi^{\pm}K^{\mp}\pi^{+}\pi^{-}]_{D}K^{\pm}$ and
$B^{\pm}\rightarrow[\pi^{\pm}K^{\mp}\pi^{+}\pi^{-}]_{D}\pi^{\pm}$ using
$1.0\,{\rm fb^{-1}}$ of data collected by LHCb in 2011. First observations
have been made of both decays, with a significance of $5.1\sigma$ and greater
than $10\sigma$, respectively. Measurements have been made of the observables
$R^{K3\pi}_{K/\pi}$, $A^{K3\pi}_{h}$ and $R_{h}^{K3\pi,\pm}$, as well as the
derived parameters $R^{K3\pi}_{{\rm ADS}(h)}$ and $A^{K3\pi}_{{\rm ADS}(h)}$,
which relate the partial widths of the $B^{\pm}\rightarrow Dh^{\pm}$
($h=K,\pi$) family of decays. From these observables it is deduced that
$r_{B}^{K}=0.097\pm 0.011$, where $r_{B}^{K}$ is the ratio of the absolute
values of the suppressed and favoured $B^{\pm}\rightarrow DK^{\pm}$
amplitudes. These results will improve knowledge of the Unitarity Triangle
angle $\gamma$ when they are combined with other $B^{\pm}\rightarrow DK^{\pm}$
measurements exploiting different $D$ decay modes.

## 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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# Forecasting COVID-19 Pandemic in Mozambique and Estimating Possible
Scenarios

Cláudio Moisés Paulo
Grupo de Astrofísica e Ciências Espaciais
Universidade Eduardo Mondlane
Av. Julius Nyerere, nº 3453,Campus Principal
<EMAIL_ADDRESS>
&Felipe Nunes Fontinele
Department of Physics, Faculty of Sciences
University of Alberta
4-181 CCIS, Edmonton, Alberta, Canada T6G 2E1
<EMAIL_ADDRESS>
&Pedro Henrique P. Cintra
Instituto de Física
Universidade de Brasília
Campus Darcy Ribeiro, Asa norte, Brasília, Brazil
<EMAIL_ADDRESS>

###### Abstract

COVID-19 is now the largest pandemic crisis of this century, with over 16
million registered cases worldwide. African countries have now begun
registering an increasing number of cases, yet, not many models developed
focus in specific African countries. In our study we use a simple SEIR model
to evaluate and predict future scenarios regarding the pandemic crisis in
Mozambique. We compare the effect of different policies on the infection curve
and estimate epidemiological parameters such as the current infection
reproduction number $\mathcal{R}_{t}$ and the growth rate $g$. We have found a
low value for $\mathcal{R}_{t}$, ranging from 1.11 to 1.48 and a positive
growth rate, between $g$ = 0.22 to 0.27. Our simulations also suggest that a
lockdown shows potential for reducing the infection peak height in 28%, on
average, ranging from 20 to 36%.

_K_ eywords SEIR $\cdot$ Mozambique $\cdot$ COVID-19 $\cdot$ Forecast $\cdot$
Coronavirus

##  1 Introduction

The ongoing Coronavirus disease 2019 (COVID-19) pandemic crisis has now over
16 million registered cases worldwide and more than 600 thousand deaths,
according to the situation report 186 by the World Health Organization
(https://www.who.int/emergencies/diseases/novel-coronavirus-2019/situation-
reports). COVID-19 is a respiratory disease caused by the SARS-CoV-2 virus,
novel species of the betacoronavirus genus, it shows high affinity with
Angiotensin converting enzyme 2 (ACE2) receptor in human cells [1, 2, 3]. The
most probable origin of the virus was traced back to bat populations by
genetic studies [4, 5], with the Pangolin being a probable intermediate host
[6].

Nowadays, the disease presents a growing behavior throughout the African
continent. With no recent scientific studies concerning the projection of the
pandemic dynamics for Mozambique, local authorities must rely on international
measures in order to control the spreading rate of COVID-19. This study aims
at accessing the problem in Mozambique, estimating important epidemiological
factors such as the growth rate $g$ and the current value of the infection
reproduction number $\mathcal{R}_{t}$. Using a simple SEIR model including the
dead compartment, we rely on daily epidemiological information available by
the government (https://covid19.ins.gov.mz/documentos/); concerning the
cumulative number of infections until July 24, cumulative number of recoveries
and deaths, to forecast the possible futures the pandemic might take on the
country.

Mathematical models are now worldwide used for forecasting [7, 8, 9, 10], data
analysis [11] and economic impact [12, 13] regarding the current pandemic
crisis. SIR and SEIR models are the most simple and used ones for evaluation
of the outbreak dynamics [14]. Although more complex models are frequently
employed and might yield more specific results; such as models concerning
quarantine, asymptomatic and hospitalizations, these models make use of many
unknown parameters, presenting a challenge when trying to copy reality in
simulations.

##  2 Model Description

We apply a modified SEIR model to include deaths as a new compartment. The
model relies on the following chain of events: A susceptible individual from a
population $N$ of constant size is exposed to the virus, joining the exposed
group, in which the person does not show symptoms of the disease yet, after
some time (incubation period) this exposed person joins the infected group,
where the symptoms onset. From this stage the patient either recovers or dies.

The model treats of populations, therefore these transitions between groups
are given by rates of changes in a specific population. Susceptible
individuals, denoted by $S$, become infected through contact with infected
($I$) or exposed ($E$) individuals at a rate proportional to the density of
infected and exposed $[(1-P_{exp})\beta I+P_{exp}\beta E]/N$, where $P_{exp}$
is the percentage of infections caused by the exposed population and $\beta$
is the infection rate. The exposed population declines as patients become
infected by a rate $c$ proportional to the inversion of the incubation period
$c=1/\tau$, with $\tau$ being the incubation period. Once inside the infected
population, patients now transit to recovered ($R$) or dead ($D$) groups
through constant rates $\gamma$ and $\mu$ respectively. These rates are
proportional to the infection fatality rate (IFR) and the average time taken
from symptoms onset to recovery $\tau_{r}$ and the average time taken from
symptoms onset to death $\tau_{d}$ (Figure 1).

Figure 1: Representation of the SEIRD model described above.

The dynamics described by these populations is mathematically represented by
the following set of differential equations

$\displaystyle\frac{dS}{dt}=-\frac{(1-P_{exp})\beta}{N}IS-\frac{P_{exp}\beta}{N}ES$
(1)
$\displaystyle\frac{dE}{dt}=\frac{(1-P_{exp})\beta}{N}IS+\frac{P_{exp}\beta}{N}E-cE$
(2) $\displaystyle\frac{dI}{dt}=cE-(\gamma+\mu)I$ (3)
$\displaystyle\frac{dR}{dt}=\gamma I$ (4) $\displaystyle\frac{dD}{dt}=\mu I$
(5)

where $\gamma$ and $\mu$ are given by

$\displaystyle\gamma=\frac{(1-IFR)}{\tau_{r}}$ (6)
$\displaystyle\mu=\frac{IFR}{\tau_{d}}.$

These equations are subjected to a set of initial conditions at $t=0$; here we
assume $S(t=0)=S_{0}\approx N$, $E(t=0)=E_{0}$, $I(t=0)=I_{0}$, $R(t=0)=0$ and
$D(t=0)=0$. An important consideration taken into account by the model is that
the total population $N$ is constant, which can be noted by summing equations
(1) - (5) and noticing that it equals to 0, therefore
$dN/dt=dS/dt+dE/dt+dI/dt+dR/dt+dD/dt=0\Rightarrow N=\text{constant}$. From
this type of model we retrieve the basic reproduction number, from which we
calculate the current infection reproduction number using the current value
for $\beta$ instead of the natural value without intervention policies, for
the SEIRD model which was already derived in a previous work [15], where was
made use of a method developed in [16], resulting in

$\displaystyle\mathcal{R}_{0}=\frac{P_{exp}\beta\left[\gamma+(1-P_{exp})\beta\right]+(1-P_{exp})\beta
c}{c(\gamma+\mu)}$ (7)

Another important quantity derived from the model and widely used to access
the growth of the outbreak in a particular region is the growth rate, given by

$\displaystyle g=\beta-(\gamma+\mu)$ (8)

which states the difference between new infections and new outcomes from
previous infections.

###  2.1 Non-Pharmaceutical Intervention

The next issue treated by our model is the description of non-pharmaceutical
interventions such as social isolation and lockdown. We assume these actions
alter the infection rate as time progresses; the core of the infection rate
$\beta$ embeds the average daily contacts an individual has and the
probability of being infected at each contact. Therefore, we consider a time-
variant $\beta$ obeying a logistic behavior described by the following
equation

$\displaystyle\frac{\beta(t)}{N}=\frac{P_{red}\beta_{i}/N}{1+\Gamma
e^{b(t-t_{c})}}+(1-P_{red})\beta_{i}/N$ (9)

where $P_{red}$ is the percentage of reduction suffered by $\beta(t)$ as a
result from the intervention, $\beta_{i}$ is the initial value $\beta(t)$,
that is, the value $\beta(t)$ would acquire without the intervention.
Parameters $t_{c}$, $b$ and $\Gamma$ are, respectively, the critical time when
the intervention starts, the control on how fast $\beta$ decreases to the new
value and a scale factor adjusted so the decrease of $\beta(t)$ starts at the
given date; here we choose $\Gamma=0.01$ and $b=1$, with those parameters set
the time taken for $\beta(t)$ do decrease the desirable amount represented by
$P_{red}$ would be approximately 10 days.

Considering this behavior we may represent the action of different non-
pharmaceutical interventions based on their effectiveness and starting date.
We might also add more logistic functions with different parameters and
different sign for $b$, in such a way to predict the ending of a given
intervention policy.

##  3 Simulations

In order to simulate future scenarios and the actual tendency of Mozambique
regarding the dynamics of the outbreak, we used a Python code freely available
at https://github.com/PedroHPCintra/COVID-Mozambique. Using the non-linear
least squares method, we fitted infected data constructed from the subtraction
of the cumulative number by the recoveries and deaths ($I=CT-R-D$, where $CT$
is the cumulative total), from the fit we acquired values of epidemiological
parameter, allowing the acquisition of the growth rate $g$ and the infection
reproduction number $\mathcal{R}_{t}$. After fitting the data, we used the
fitted parameters to simulate the future tendency of behavior, as well as
future scenarios developed in section 4.1.

For the fitting we decided to leave the parameters $\beta$, $E_{0}$ and
$I_{0}$ free, while $\gamma$, $\mu$, $c$, $t_{c}$, $P_{red}$ are given in
table 1. The major problem with fitting data at initial stages of the pandemic
is the computational estimation of $S_{0}$, which in our case is approximated
to the total population $N$. The mathematical model presumes a homogeneously
distributed population throughout the country, which is not a considerable
reality, as shown in figure 2, thus, we cannot consider $N$ as being the total
population of Mozambique; instead, $N$ plays the role of an effective
population, which is the equivalent population of the country if the spatial
distribution of people was indeed homogeneous.

Figure 2: Demographic density of Mozambique. From the map it is possible to
verify the higher demographic concentration in the region of Nampula and
Zambezia, as well as localized districts of higher population per $Km^{2}$.

Aiming to solve this limitation, we ran several simulations using different
fixed values of $N$, ranging from $0.01\%$ of the population of Cabo Delgado +
Maputo + Nampula to $10\%$. The choice of these regions to represent the
effective $N$ of Mozambique lies in the fact that they represent the vast
majority of cases, that way, we approximate the population of Mozambique to be
fractions of the populations of Nampula, Cabo Delgado and Maputo. At each
stage, the $\chi^{2}$ of the fit was taken into account and, if it was lower
than 0.98, the fit was discarded. In this way, we narrow the range of possible
$N$s that might describe this first pandemic wave. We discuss in more detail
how $N$ changes during the pandemic in section 5.

Parameter | Value | Reference
---|---|---
$IFR$ | 0.5% | [17]
$c$ | 5.1 days-1 | [18, 19, 20, 21]
$\tau_{r}$ | 10 days | [22]
$\tau_{d}$ | 18 days | [23]
$t_{c}$ | 8 days | Local government decree
$P_{red}$ | 50% | [24]
$P_{exp}$ | 44% | [25]
Table 1: Parameters used for the simulation and the fitting of epidemiological
data from Mozambique.

Later, when simulating the possible scenarios in section 4.1, we added a
second logistic drop to $\beta(t)$ starting at the supposed date for a
lockdown imposition, at the beginning of August. The drop decreased
$\beta_{i}$ 20% more, that way the total decrease suffered by $\beta_{i}$
would be 70%, which is consistent with the efficacy of decrease lockdowns
acquire, supposing that the decrease in $\mathcal{R}_{t}$ 111$\mathcal{R}_{t}$
represents the change $\mathcal{R}_{0}$ suffers due to intervention policies
at the outbreak. While $\mathcal{R}_{0}$ is specific for each disease and
represents the spread without any intervention, $\mathcal{R}_{t}$ changes with
time due to intervention policies. or in the growth rate $g$ is due to a
decrease in the infection rate $\beta$ [17]. Then, to simulate faster or
slower re-openings, we considered the scenarios where the infection rate grows
as fast as it decreased (fast re-opening) and the scenario where the infection
rate grows 10 times slower than it decreased (slow re-opening); that is done
by changing the $b$ parameter in a third logistic function with inverted sign
at $b$, representing a increase, instead of a drop.

Figure 3: Representation of the temporal change in $\beta(t)$ due to
intervention policies. Here we use a sum of three logistic curves with
$t_{c}$s corresponding to the date in which each policy is applied. The $b$
parameter for the third logistic function, representing the return to daily
activities, was chosen to be -0.1, in contrast to $b=1$ at the first two
logistic curves.

##  4 Results

When performing the fitting, the minimum value of $N$ acceptable by the
$\chi^{2}$ elimination was $N_{min}=0.09\%$ of the population, while the
largest value was $N_{max}=0.5\%$ of the population. We then performed
fittings considering $N=0.09\%$, $0.1\%$, $0.2\%$, $0.3\%$, $0.4\%$ and
$0.5\%$ and $\beta_{i}=0.4$, $I_{0}=1$ and $E_{0}=5$ as a initial guess;
$\beta_{i}$ was chosen according to the observed value when fitting data from
other countries with reliable data [26]. Our first adjustment of the current
data yielded $\beta_{i}=0.31\pm 0.02$, $I_{0}=2\pm 1$ and $E_{0}=7\pm 5$.

Having $\beta_{i}$ at hand, we calculated the range of $g$ and
$\mathcal{R}_{t}$ in Mozambique, using the known international values for
$\gamma$, $\mu$, $c$ and $P_{exp}$ given by table 1.

$\displaystyle g=0.25\pm 0.03$ (10) $\displaystyle\mathcal{R}_{t}=1.29\pm
0.19$ (11)

The results of the possible tendencies for the future behavior of the
pandemic, according to the recent data, is shown at figure 4.

Figure 4: Predictions of possible behaviors of the outbreak taking into
account the current data available. The color gradient represents each $N$
simulated, blue curve is the simulated scenario with $N=0.09\%$, cyan region
represents $N=0.1\%$, green $N=0.2\%$, yellow = $N=0.3\%$ and so on. Infection
peak is estimated to be between the final of August and September, with a
infection peak height between 1500 infections (on the best scenario) and 3800
(on the worse scenario)

The current scenario predicted to Mozambique shows that the country might be
close to an infection peak, given the current tendency; however it is also
possible for the infection curve to increase at least for two more months and
reach an infection peak close to 3500 cases, with a cumulative number of cases
around 30000 after 400 days of pandemic.

###  4.1 Future Scenarios

After the adjustment, we considered some other possible scenarios for the
future behavior of the outbreak. The first was considering a national lockdown
taking place at the beginning of August (Figure 7), the second was simulating
a full re-opening of commerce, schools and daily life return in 10 days after
the first day of August (Figure 7), the third scenario was a slower re-
opening, 10 times slower than the previous case (Figure 7). The aim when
considering these projections is to provide useful information for policy
management when thinking about the future dynamics of the pandemic crisis.

According to these projections the return to daily activities on August could
increase the total number of COVID-19 infections at the peak 3.4 times,
depending on the effective $N$, ranging from a 1.2 increase in peak height to
3.4. By the other hand, a slower return, according to the consideration in
which the infection rate $\beta$ increases 10 times slower, could provoke an
increase of the peak number from 0 to 2.3. Finally, the lockdown may decrease
the infection peak to 60% of the current maximum tendency, which is
represented by the red curve in figure 7. At the best scenario, represented by
the blue and cyan curves in figure 7, we see that the lockdown would not
affect the peak height, however it would result in a faster decrease of the
infection curve after the peak.

As expected, a national lockdown cannot be held forever until the last
infection registered due to economical costs. We then, estimated 6 scenarios
for duration of a lockdown and re-opening; corresponding to a lockdown for 30,
60 and 90 days and the same situation considering 30, 60 and 90 days of
lockdown with a slow return to daily activities (Figures 8(a) to 8(f)).

The projections show that a slower re-opening might decrease the second peak
height by 36%, considering a lockdown that lasts 30 days, 29%, for 60 days
duration, and 20% for a 90 days duration. On average, the gradual return to
daily life shows the potential to decrease the second wave intensity by 28%.

Considering the duration of the lockdown, the longer the duration time for the
intervention, the lower the second peak is after return.

Figure 5: Predictions for the future behavior of the outbreak considering a
lockdown with indefinite duration starting at the beginning of August, that
is, close to the 130th day after the first case.

Figure 6: Predictions for the future behavior of the outbreak considering a
re-opening of schools and commerce at the beginning of August, close to the
130th day after the first case.

Figure 7: Predictions for the future behavior of the outbreak considering a 10
times slower re-opening of schools and commerce at the beginning of August,
close to the 130th day after the first case.

(a) Predictions considering a lockdown of 30 days with a fast return to daily
life.

(b) Predictions considering a lockdown of 30 days with a slow return to daily
life.

(c) Predictions considering a lockdown of 60 days with a fast return to daily
life.

(d) Predictions considering a lockdown of 60 days with a slow return to daily
life.

(e) Predictions considering a lockdown of 90 days with a fast return to daily
life.

(f) Predictions considering a lockdown of 90 days with a slow return to daily
life.

Figure 8: The color gradient represents each $N$ simulated, blue curve is the
simulated scenario with $N=0.09\%$, cyan region represents $N=0.1\%$, green
$N=0.2\%$, yellow = $N=0.3\%$ and so on. Figures 8(a) and 8(b) represent the
behavior for a lockdown lasting 30 days, taking place at the beggining of
August, at the 130th day of pandemic. They both represent the scenarios where
the return to normal activies in commerce, schools and home isolation, is
taken fast (8(a)) and slow (8(b)). Figures 8(c) and 8(d) represent the same
scenario, however with a longer duration for the lockdown; 60 days. Finally,
figures 8(e) and 8(f) shows the final scenario where the lockdown lasts 90
days.

##  5 Discussion

Our aim in this study is to evaluate the future dynamics of COVID-19 pandemic
crisis in Mozambique. We have shown that simulations predict over 200%
increase of peak intensity if commercial activities and daily life returns to
normal in August, when compared to the tendency of cases. Moreover, lockdown
policies and slower returns substantially decrease the peak intensity.

The main limitation of our study is the effective population $N$ susceptible
to the disease. In the real situation, $N$ increases with time, as the disease
progresses and the virus begging circulating in more interior regions. That
behavior could provoke an increase in the susceptible population, causing more
abrupt peaks and creating a bigger number of cumulative cases. Therefore, our
estimations here should not be taken as a precise prediction regarding the
numbers of infections, instead, we draw different scenarios and show how
different policies might change the infection curve, given the recent data for
Mozambique.

Another limitation lies on the data, which might be biased by a lower testing
rate. However, to date, Mozambique performed 31.7 tests for each positive
case, on average, which is close to Spanish numbers according to the online
platform Ourworldindata (https://ourworldindata.org/coronavirus-testing). In
countries known to have acquired a accurate tracking of cases, such as
Germany, South Korea and Australia, these numbers jump to 191.9, 201.5 and
196.8, respectively. By the other hand, countries known to have a poor
accuracy in tracking of cases such as Brazil and United States have 1.5 and
11.5 tests per confirmed case, respectively. We could estimated that
Mozambique data is not perfect and might lose some cases, provoking changes on
the future of the pandemic, however it is not poorly tracked.

The last limitation lies in the model and the algorithm itself; first, the
model does not take into account infections caused by contact infected
surfaces, which is known to be a viable source of infection due to viral
persistence at inanimate surfaces [27]. Second, the algorithm used is simple
and direct, but is susceptible to the nonidentifiability problem [28], which
arises in non statistical fitting methods. However this is not expected to
change drastically the shape of the curve.

Our study also found a positive growth rate, indication of a continuous growth
of cases in Mozambique. The $\mathcal{R}_{t}$ number is however low,
suggesting a slow increase of infection curve. However, due to the lower
number of infections, to date, Mozambique might still be on a stochastic limit
of behavior, making the measurement of $\mathcal{R}_{t}$ imprecise, but not
very different from the value found here.

The infection curve might be close to the infection peak, considering the best
scenario, however it is also possible to observe a prolonged increase, as the
worse scenario curves have shown. It is not possible to say which curve the
country will tend to follow, this mainly depends on the effective population
in reach of the disease. Therefore, we highlight that our study represents
simple views of the real scenario and should not represent the precise future
of Mozambique, instead it should provide mere estimates for the possible
behaviors.

## Conflict of Interest

The authors declare no conflict of interest.

## Funding

The study received no external funding or resources.

## Acknowledgments

Our thanks to Francesse Mauro and Márcio Mathe, from the Department of
Mathematics, course of Geographical Information System, in the Universidade
Eduardo Mondlane, to providing us the map for the demographic density of
Mozambique.

Cláudio Moisés Paulo thanks to the project Development in Africa with Radio
Astronomy (DARA, https://www.dara-project.org/) and to the project Development
of Palop Knowledge in Radioastronomy (DOPPLER222DOPPLER is funded by the Aga
Khan Development Network and the Fundação para a Ciência e a Tecnologia (FCT),
under grant number 333197717., http://doppler.av.it.pt/), for providing him
equipment which was used for this study.

## Additional Information

Our source code is freely available at https://github.com/PedroHPCintra/COVID-
Mozambique.

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