EUROPEAN ORGANISATION FOR NUCLEAR RESEARCH (CERN)

 CERN-PH-EP-2013-079 LHCb-PAPER-2013-020 24\. July 2013

Measurement of the CKM angle $\gamma$ from a combination of
$B^{\pm}\\!\rightarrow Dh^{\pm}$ analyses

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

A combination of three LHCb measurements of the CKM angle $\gamma$ is
presented. The decays $B^{\pm}\\!\rightarrow DK^{\pm}$ and
$B^{\pm}\\!\rightarrow D\pi^{\pm}$ are used, where $D$ denotes an admixture of
$D^{0}$ and $\kern 1.99997pt\overline{\kern-1.99997ptD}{}^{0}$ mesons,
decaying into $K^{+}K^{-}$, $\pi^{+}\pi^{-}$, $K^{\pm}\pi^{\mp}$,
$K^{\pm}\pi^{\mp}\pi^{\pm}\pi^{\mp}$, $K^{0}_{\rm\scriptscriptstyle
S}\pi^{+}\pi^{-}$, or $K^{0}_{\rm\scriptscriptstyle S}K^{+}K^{-}$ final
states. All measurements use a dataset corresponding to $1.0\mbox{\,fb}^{-1}$
of integrated luminosity. Combining results from $B^{\pm}\\!\rightarrow
DK^{\pm}$ decays alone a best-fit value of $\gamma=72.0^{\circ}$ is found, and
confidence intervals are set

$\displaystyle\gamma\in[56.4,86.7]^{\circ}\quad$ $\displaystyle{\rm at\
68\%\,CL}\,,$ $\displaystyle\gamma\in[42.6,99.6]^{\circ}\quad$
$\displaystyle{\rm at\ 95\%\,CL}\,.$

The best-fit value of $\gamma$ found from a combination of results from
$B^{\pm}\\!\rightarrow D\pi^{\pm}$ decays alone, is $\gamma=18.9^{\circ}$, and
the confidence intervals

$\displaystyle\gamma\in[7.4,99.2]^{\circ}\quad\cup\quad[167.9,176.4]^{\circ}\quad$
$\displaystyle{\rm at\ 68\%\,CL}\,$

are set, without constraint at $95\%$ CL. The combination of results from
$B^{\pm}\\!\rightarrow DK^{\pm}$ and $B^{\pm}\\!\rightarrow D\pi^{\pm}$ decays
gives a best-fit value of $\gamma=72.6^{\circ}$ and the confidence intervals

$\displaystyle\gamma\in[55.4,82.3]^{\circ}\quad$ $\displaystyle{\rm at\
68\%\,CL}\,,$ $\displaystyle\gamma\in[40.2,92.7]^{\circ}\quad$
$\displaystyle{\rm at\ 95\%\,CL}\,$

are set. All values are expressed modulo $180^{\circ}$, and are obtained
taking into account the effect of $D^{0}$–$\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}$ mixing.

To be 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. Andreassen56, R.B.
Appleby53, O. Aquines Gutierrez10, F. Archilli18, A. Artamonov34, M. Artuso58,
E. Aslanides6, G. Auriemma24,m, S. Bachmann11, J.J. Back47, C. Baesso59, V.
Balagura30, W. Baldini16, R.J. Barlow53, C. Barschel37, S. Barsuk7, W.
Barter46, Th. Bauer40, A. Bay38, J. Beddow50, F. Bedeschi22, I. Bediaga1, S.
Belogurov30, K. Belous34, I. Belyaev30, E. Ben-Haim8, G. Bencivenni18, S.
Benson49, J. Benton45, A. Berezhnoy31, R. Bernet39, M.-O. Bettler46, M. van
Beuzekom40, A. Bien11, S. Bifani44, T. Bird53, A. Bizzeti17,h, P.M.
Bjørnstad53, T. Blake37, F. Blanc38, J. Blouw11, S. Blusk58, V. Bocci24, A.
Bondar33, N. Bondar29, W. Bonivento15, S. Borghi53, A. Borgia58, T.J.V.
Bowcock51, E. Bowen39, C. Bozzi16, T. Brambach9, J. van den Brand41, J.
Bressieux38, D. Brett53, M. Britsch10, T. Britton58, N.H. Brook45, H. Brown51,
I. Burducea28, A. Bursche39, G. Busetto21,p, J. Buytaert37, S. Cadeddu15, O.
Callot7, M. Calvi20,j, M. Calvo Gomez35,n, A. Camboni35, P. Campana18,37, D.
Campora Perez37, A. Carbone14,c, G. Carboni23,k, R. Cardinale19,i, A.
Cardini15, H. Carranza-Mejia49, L. Carson52, K. Carvalho Akiba2, G. Casse51,
L. Castillo Garcia37, M. Cattaneo37, Ch. Cauet9, M. Charles54, Ph.
Charpentier37, P. Chen3,38, N. Chiapolini39, M. Chrzaszcz39, K. Ciba37, X. Cid
Vidal37, G. Ciezarek52, P.E.L. Clarke49, M. Clemencic37, H.V. Cliff46, J.
Closier37, C. Coca28, V. Coco40, J. Cogan6, E. Cogneras5, P. Collins37, A.
Comerma-Montells35, A. Contu15, A. Cook45, M. Coombes45, S. Coquereau8, G.
Corti37, B. Couturier37, G.A. Cowan49, D.C. Craik47, S. Cunliffe52, R.
Currie49, C. D’Ambrosio37, P. David8, P.N.Y. David40, A. Davis56, I. De
Bonis4, K. De Bruyn40, S. De Capua53, M. De Cian39, J.M. De Miranda1, L. De
Paula2, W. De Silva56, P. De Simone18, D. Decamp4, M. Deckenhoff9, L. Del
Buono8, N. Déléage4, D. Derkach14, O. Deschamps5, F. Dettori41, A. Di Canto11,
H. Dijkstra37, M. Dogaru28, S. Donleavy51, F. Dordei11, A. Dosil Suárez36, D.
Dossett47, A. Dovbnya42, F. Dupertuis38, R. Dzhelyadin34, A. Dziurda25, A.
Dzyuba29, S. Easo48,37, U. Egede52, V. Egorychev30, S. Eidelman33, D. van
Eijk40, S. Eisenhardt49, U. Eitschberger9, R. Ekelhof9, L. Eklund50,37, I. El
Rifai5, Ch. Elsasser39, D. Elsby44, A. Falabella14,e, C. Färber11, G.
Fardell49, C. Farinelli40, S. Farry51, V. Fave38, D. Ferguson49, V. Fernandez
Albor36, F. Ferreira Rodrigues1, M. Ferro-Luzzi37, S. Filippov32, M. Fiore16,
C. Fitzpatrick37, M. Fontana10, F. Fontanelli19,i, R. Forty37, O. Francisco2,
M. Frank37, C. Frei37, M. Frosini17,f, S. Furcas20, E. Furfaro23,k, A. Gallas
Torreira36, D. Galli14,c, M. Gandelman2, P. Gandini58, Y. Gao3, J. Garofoli58,
P. Garosi53, J. Garra Tico46, L. Garrido35, C. Gaspar37, R. Gauld54, E.
Gersabeck11, M. Gersabeck53, T. Gershon47,37, Ph. Ghez4, V. Gibson46, V.V.
Gligorov37, C. Göbel59, D. Golubkov30, A. Golutvin52,30,37, A. Gomes2, H.
Gordon54, M. Grabalosa Gándara5, R. Graciani Diaz35, L.A. Granado Cardoso37,
E. Graugés35, G. Graziani17, A. Grecu28, E. Greening54, S. Gregson46, P.
Griffith44, O. Grünberg60, B. Gui58, E. Gushchin32, Yu. Guz34,37, T. Gys37, C.
Hadjivasiliou58, G. Haefeli38, C. Haen37, S.C. Haines46, S. Hall52, T.
Hampson45, S. Hansmann-Menzemer11, N. Harnew54, S.T. Harnew45, J. Harrison53,
T. Hartmann60, J. He37, V. Heijne40, K. Hennessy51, P. Henrard5, J.A. Hernando
Morata36, E. van Herwijnen37, A. Hicheur1, E. Hicks51, D. Hill54, M.
Hoballah5, C. Hombach53, P. Hopchev4, W. Hulsbergen40, P. Hunt54, T. Huse51,
N. Hussain54, D. Hutchcroft51, D. Hynds50, V. Iakovenko43, M. Idzik26, P.
Ilten12, R. Jacobsson37, A. Jaeger11, E. Jans40, P. Jaton38, A. Jawahery57, F.
Jing3, M. John54, D. Johnson54, C.R. Jones46, C. Joram37, B. Jost37, M.
Kaballo9, S. Kandybei42, M. Karacson37, T.M. Karbach37, I.R. Kenyon44, U.
Kerzel37, T. Ketel41, A. Keune38, B. Khanji20, O. Kochebina7, I. Komarov38,
R.F. Koopman41, P. Koppenburg40, M. Korolev31, A. Kozlinskiy40, L. Kravchuk32,
K. Kreplin11, M. Kreps47, G. Krocker11, P. Krokovny33, F. Kruse9, M.
Kucharczyk20,25,j, V. Kudryavtsev33, T. Kvaratskheliya30,37, V.N. La Thi38, D.
Lacarrere37, G. Lafferty53, A. Lai15, D. Lambert49, R.W. Lambert41, E.
Lanciotti37, G. Lanfranchi18,37, C. Langenbruch37, T. Latham47, C.
Lazzeroni44, R. Le Gac6, J. van Leerdam40, J.-P. Lees4, R. Lefèvre5, A.
Leflat31, J. Lefrançois7, S. Leo22, O. Leroy6, T. Lesiak25, B. Leverington11,
Y. Li3, L. Li Gioi5, M. Liles51, R. Lindner37, C. Linn11, B. Liu3, G. Liu37,
S. Lohn37, I. Longstaff50, J.H. Lopes2, E. Lopez Asamar35, N. Lopez-March38,
H. Lu3, D. Lucchesi21,p, 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, 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, B. Mc Skelly51, J. McCarthy44, A. McNab53, R.
McNulty12, B. Meadows56,54, F. Meier9, M. Meissner11, M. Merk40, D.A.
Milanes8, M.-N. Minard4, J. Molina Rodriguez59, S. Monteil5, D. Moran53, P.
Morawski25, M.J. Morello22,r, R. Mountain58, I. Mous40, F. Muheim49, K.
Müller39, R. Muresan28, B. Muryn26, B. Muster38, P. Naik45, T. Nakada38, R.
Nandakumar48, I. Nasteva1, M. Needham49, N. Neufeld37, A.D. Nguyen38, T.D.
Nguyen38, C. Nguyen-Mau38,o, M. Nicol7, V. Niess5, R. Niet9, N. Nikitin31, T.
Nikodem11, A. Nomerotski54, A. Novoselov34, A. Oblakowska-Mucha26, V.
Obraztsov34, S. Oggero40, S. Ogilvy50, O. Okhrimenko43, R. Oldeman15,d, M.
Orlandea28, J.M. Otalora Goicochea2, P. Owen52, A. Oyanguren35, B.K. Pal58, A.
Palano13,b, M. Palutan18, J. Panman37, A. Papanestis48, M. Pappagallo50, C.
Parkes53, C.J. Parkinson52, G. Passaleva17, G.D. Patel51, M. Patel52, G.N.
Patrick48, C. Patrignani19,i, C. Pavel-Nicorescu28, A. Pazos Alvarez36, A.
Pellegrino40, G. Penso24,l, M. Pepe Altarelli37, S. Perazzini14,c, D.L.
Perego20,j, E. Perez Trigo36, A. Pérez-Calero Yzquierdo35, P. Perret5, M.
Perrin-Terrin6, G. Pessina20, K. Petridis52, A. Petrolini19,i, A. Phan58, E.
Picatoste Olloqui35, B. Pietrzyk4, T. Pilař47, D. Pinci24, S. Playfer49, M.
Plo Casasus36, F. Polci8, G. Polok25, A. Poluektov47,33, E. Polycarpo2, A.
Popov34, D. Popov10, B. Popovici28, C. Potterat35, A. Powell54, J.
Prisciandaro38, A. Pritchard51, C. Prouve7, V. Pugatch43, A. Puig Navarro38,
G. Punzi22,q, W. Qian4, J.H. Rademacker45, B. Rakotomiaramanana38, M. Rama18,
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, G. Sabatino24,k, J.J. Saborido Silva36,
N. Sagidova29, P. Sail50, B. Saitta15,d, V. Salustino Guimaraes2, C.
Salzmann39, B. Sanmartin Sedes36, M. Sannino19,i, R. Santacesaria24, C.
Santamarina Rios36, E. Santovetti23,k, M. Sapunov6, A. Sarti18,l, C.
Satriano24,m, A. Satta23, M. Savrie16,e, D. Savrina30,31, P. Schaack52, M.
Schiller41, H. Schindler37, M. Schlupp9, M. Schmelling10, B. Schmidt37, O.
Schneider38, A. Schopper37, M.-H. Schune7, R. Schwemmer37, B. Sciascia18, A.
Sciubba24, M. Seco36, A. Semennikov30, K. Senderowska26, I. Sepp52, N.
Serra39, J. Serrano6, P. Seyfert11, M. Shapkin34, I. Shapoval16,42, P.
Shatalov30, Y. Shcheglov29, T. Shears51,37, L. Shekhtman33, O. Shevchenko42,
V. Shevchenko30, A. Shires52, R. Silva Coutinho47, T. Skwarnicki58, N.A.
Smith51, E. Smith54,48, M. Smith53, M.D. Sokoloff56, F.J.P. Soler50, F.
Soomro18, D. Souza45, B. Souza De Paula2, B. Spaan9, A. Sparkes49, P.
Spradlin50, F. Stagni37, S. Stahl11, O. Steinkamp39, S. Stoica28, S. Stone58,
B. Storaci39, M. Straticiuc28, U. Straumann39, V.K. Subbiah37, L. Sun56, S.
Swientek9, V. Syropoulos41, M. Szczekowski27, P. Szczypka38,37, T. Szumlak26,
S. T’Jampens4, M. Teklishyn7, E. Teodorescu28, F. Teubert37, C. Thomas54, E.
Thomas37, J. van Tilburg11, V. Tisserand4, M. Tobin38, S. Tolk41, D.
Tonelli37, S. Topp-Joergensen54, N. Torr54, E. Tournefier4,52, S. Tourneur38,
M.T. Tran38, M. Tresch39, A. Tsaregorodtsev6, P. Tsopelas40, N. Tuning40, M.
Ubeda Garcia37, A. Ukleja27, D. Urner53, U. Uwer11, V. Vagnoni14, G.
Valenti14, R. Vazquez Gomez35, P. Vazquez Regueiro36, S. Vecchi16, J.J.
Velthuis45, M. Veltri17,g, G. Veneziano38, M. Vesterinen37, B. Viaud7, D.
Vieira2, X. Vilasis-Cardona35,n, A. Vollhardt39, D. Volyanskyy10, D. Voong45,
A. Vorobyev29, V. Vorobyev33, C. Voß60, H. Voss10, R. Waldi60, R. Wallace12,
S. Wandernoth11, J. Wang58, D.R. Ward46, N.K. Watson44, A.D. Webber53, D.
Websdale52, M. Whitehead47, J. Wicht37, J. Wiechczynski25, D. Wiedner11, L.
Wiggers40, G. Wilkinson54, M.P. Williams47,48, M. Williams55, F.F. Wilson48,
J. Wishahi9, M. Witek25, S.A. Wotton46, S. Wright46, S. Wu3, K. Wyllie37, Y.
Xie49,37, Z. Xing58, Z. Yang3, R. Young49, X. Yuan3, O. Yushchenko34, M.
Zangoli14, M. Zavertyaev10,a, F. Zhang3, L. Zhang58, W.C. Zhang12, Y. Zhang3,
A. Zhelezov11, A. Zhokhov30, L. Zhong3, A. Zvyagin37.

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

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

##  1 Introduction

The angle $\gamma$ is defined as
$\gamma=\arg\left[-V^{\phantom{*}}_{ud}V^{*}_{ub}/(V^{\phantom{*}}_{cd}V^{*}_{cb})\right]$,
where $V_{ij}$ are the elements of the Cabibbo-Kobayashi-Maskawa (CKM) matrix
[1, *Kobayashi:1973fv]. It is one of the angles of the unitarity triangle and
is to date the least well known angle of this triangle. At the same time it is
the only angle that can be measured entirely with decays that only involve
tree diagrams, so its measurement is largely unaffected by the theoretical
uncertainty, which is $\mathcal{O}(10^{-6})$ [3]. Both Belle and BaBar have
recently published averages of their measurements, each following a
frequentist treatment. Belle measures $\gamma=(68^{+15}_{-14})^{\circ}$ [4],
and BaBar measures $\gamma=(69^{+17}_{-16})^{\circ}$ [5]. In this work a
combination of LHCb measurements is presented. World averages have been
computed by the CKMfitter and UTfit groups, who obtain $\gamma=(66\pm
12)^{\circ}$ [6], and $\gamma=(70.8\pm 7.8)^{\circ}$ [7], using a frequentist
and Bayesian treatment, respectively. These averages are dominated by
measurements performed at the $B$ factories, and part of all LHCb measurements
combined in this work are already included.

When measuring $\gamma$ in tree decays, an important channel is the
$B^{\pm}\\!\rightarrow DK^{\pm}$ mode, where the symbol $D$ denotes an
admixture of $D^{0}$ and $\kern 1.99997pt\overline{\kern-1.99997ptD}{}^{0}$
mesons. The $D$ meson is reconstructed in a final state accessible to both
flavour states, thus exploiting interference between the $b\rightarrow
u\bar{c}s$ and $b\rightarrow c\bar{u}s$ amplitudes. Throughout this Letter,
charge conjugation applies, unless stated otherwise. The measurements are
categorised by the $D$ meson final state: $C\\!P$ eigenstates (GLW [8, 9]),
quasi-flavour-specific states (ADS [10, 11]), and self-conjugate three-body
final states (GGSZ [12]). The small theoretical uncertainty in the measurement
of $\gamma$ is obtained in these decays because all hadronic parameters are
determined from data. The amplitude ratio
$r_{B}^{K}=|A(B^{-}\\!\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}K^{-})/A(B^{-}\\!\rightarrow
D^{0}K^{-})|$, plays a crucial role as the uncertainty on $\gamma$ scales
roughly as $1/r_{B}^{K}$. It is measured to be $r_{B}^{K}\approx 0.1$ [4, 5].

Besides the $B^{\pm}\\!\rightarrow DK^{\pm}$ channel, the
$B^{\pm}\\!\rightarrow D\pi^{\pm}$ decay also exhibits some sensitivity to
$\gamma$. The theoretical framework is fully analogous to the
$B^{\pm}\\!\rightarrow DK^{\pm}$ case. However, the respective amplitude ratio
$r_{B}^{\pi}$ is expected to be an order of magnitude smaller than
$r_{B}^{K}$, limiting the sensitivity. In this Letter, information from
$B^{\pm}\\!\rightarrow D\pi^{\pm}$ decays is included in the combined
measurement of $\gamma$ for the first time. The hadronic parameters describing
the $D$ decays are determined from data. To better constrain these parameters,
measurements by CLEO are included [13], that themselves contain inputs from
the Heavy Flavour Averaging Group (HFAG).

It has been shown that the determination of $\gamma$ from
$B^{\pm}\\!\rightarrow Dh^{\pm}$ decays, where $h=\pi,K$, is affected by
$D^{0}$–$\kern 1.99997pt\overline{\kern-1.99997ptD}{}^{0}$ mixing [14, 11, 15,
16, 17]. It enters in two parts of the analysis: in the description of the $B$
decays (e.g. through the amplitude $B^{+}\rightarrow
D^{0}K^{+}\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}K^{+}\rightarrow fK^{+}$, where $f$
denotes the $D$ final state), and in the determination of the hadronic
parameters that describe the $D$ decay. Since $D$ mixing is now well
established, its effect is included in this combination; the CLEO measurement
[13] also takes it into account explicitly. The effect of $D$ mixing on the
GLW, ADS, and GGSZ analyses is reviewed in Ref. [17]: it mostly affects the
ADS analysis of $B^{\pm}\\!\rightarrow D\pi^{\pm}$ decays, due to the small
expected value of $r_{B}^{\pi}$. The ADS analysis of $B^{\pm}\\!\rightarrow
DK^{\pm}$ decays receives a shift of $|\Delta\gamma|\lesssim 1^{\circ}$ [17].
The Dalitz-model independent GGSZ analysis of $B^{\pm}\\!\rightarrow DK^{\pm}$
is affected to a negligible extent [16, 17], and the GLW analyses of
$B^{\pm}\\!\rightarrow Dh^{\pm}$ are affected at most at order of
$\mathcal{O}(r_{B}^{K}\sqrt{x_{D}^{2}+y_{D}^{2}})$ [17], where the mixing
parameters $x_{D}$ and $y_{D}$ are at the level of $10^{-2}$. Here, a $D$
mixing measurement by LHCb [18] is included, to further constrain $x_{D}$ and
$y_{D}$.

The effect of possible $C\\!P$ violation in $D$ decays to the $\pi^{+}\pi^{-}$
and $K^{+}K^{-}$ final states [19, 20] has been discussed in Refs. [21, 22,
23]. This changes the interpretation of the observables of the GLW method,
which is included as described in Sect. 2.2.

In this combination, the strategy is to maximise a total likelihood built from
the product of the probability density functions (PDFs) $f_{i}$ of
experimental observables $\vec{A}_{i}$

$\mathcal{L}(\vec{\alpha})=\prod_{i}f_{i}(\vec{A}_{i}^{\rm
obs}|\vec{\alpha})\,,$ (1)

where the $\vec{A}_{i}^{\rm obs}$ are the measured values of the observables,
and $\vec{\alpha}$ is the set of parameters. The subscript $i$ denotes the
contributing inputs, summarised in Sects. 2.2–2.4. For most of the input
measurements it is assumed that the observables follow a Gaussian distribution

$f_{i}\propto\exp\left(-\frac{1}{2}(\vec{A}_{i}(\vec{\alpha})-\vec{A}_{i}^{\rm
obs})^{T}\,V_{i}^{-1}\,(\vec{A}_{i}(\vec{\alpha})-\vec{A}_{i}^{\rm
obs})\right)\,,$ (2)

where $V_{i}$ is the experimental covariance matrix. In this combined
measurement the statistical uncertainties dominate the resulting confidence
intervals. Therefore it is assumed that the systematic fluctuations are also
Gaussian, so that $V_{i}=V_{i}^{\rm stat}+V_{i}^{\rm syst}$. Since not all
off-diagonal entries of $V_{i}^{\rm syst}$ have been published, they are
assumed to be zero in the nominal result. An overall systematic uncertainty is
estimated due to this assumption. Any other correlations across the
statistically independent input measurements are neglected. For one pair of
variables ($\kappa_{K3\pi}$, $\delta_{K3\pi}$, described in Sect. 2) that
shows highly non-Gaussian behaviour, the experimental likelihood is taken into
account. Table 1 defines all free parameters in the global fit. The amplitude
ratios are defined as those of the suppressed processes divided by the
favoured ones. Confidence intervals on $\gamma$ and the most important
hadronic parameters are set using a frequentist procedure. The statistical
coverage of this procedure is evaluated.

Table 1: Free parameters used in the combined fit. The phase differences $\delta_{K\pi}$ and $\delta_{K3\pi}$ are defined in accordance with Refs. [5, 4, 13], they are shifted by $180^{\circ}$ with respect to the HFAG. Also, $\gamma$ gains a sign for the conjugated modes, $A(B^{+}\\!\rightarrow D^{0}h^{+})/A(B^{+}\\!\rightarrow\kern 1.79997pt\overline{\kern-1.79997ptD}{}^{0}h^{+})=r_{B}^{h}e^{i(\delta_{B}^{h}+\gamma)}$, with $h=K,\pi$.  Decay | Description | Parameter
---|---|---
$B^{\pm}\\!\rightarrow Dh^{\pm}$ |  $C\\!P$-violating weak phase | $\gamma$
| $\Gamma(B^{-}\\!\rightarrow D^{0}K^{-})/\Gamma(B^{-}\\!\rightarrow D^{0}\pi^{-})$ | $R_{\rm cab}$
$B^{\pm}\\!\rightarrow D\pi^{\pm}$ | $A(B^{-}\\!\rightarrow\kern 1.99997pt\overline{\kern-1.99997ptD}{}^{0}\pi^{-})/A(B^{-}\\!\rightarrow D^{0}\pi^{-})=r_{B}^{\pi}e^{i(\delta_{B}^{\pi}-\gamma)}$ |  $r_{B}^{\pi}$, $\delta_{B}^{\pi}$
$B^{\pm}\\!\rightarrow DK^{\pm}$ | $A(B^{-}\\!\rightarrow\kern 1.99997pt\overline{\kern-1.99997ptD}{}^{0}K^{-})/A(B^{-}\\!\rightarrow D^{0}K^{-})=r_{B}^{K}e^{i(\delta_{B}^{K}-\gamma)}$ |  $r_{B}^{K}$, $\delta_{B}^{K}$
$D^{0}\rightarrow K^{\pm}\pi^{\mp}$ | $A(D^{0}\rightarrow\pi^{-}K^{+})/A(D^{0}\rightarrow K^{-}\pi^{+})=r_{K\pi}e^{-i\delta_{K\pi}}$ |  $r_{K\pi}$, $\delta_{K\pi}$
| Cabibbo-favoured rate | $\Gamma(D\rightarrow K\pi)$
$D^{0}\rightarrow K^{\pm}\pi^{\mp}\pi^{+}\pi^{-}$ | amplitude ratio and effective strong phase diff. |  $r_{K3\pi}$, $\delta_{K3\pi}$
| coherence factor | $\kappa_{K3\pi}$
| Cabibbo-favoured rate | $\Gamma(D\rightarrow K\pi\pi\pi)$
$D^{0}\rightarrow K^{+}K^{-}$ | direct $C\\!P$ asymmetry | $A_{C\\!P}^{\rm dir}(KK)$
$D^{0}\rightarrow\pi^{+}\pi^{-}$ | direct $C\\!P$ asymmetry | $A_{C\\!P}^{\rm dir}(\pi\pi)$
$D^{0}$–$\kern 1.99997pt\overline{\kern-1.99997ptD}{}^{0}$ | mixing parameters |  $x_{D}$, $y_{D}$

##  2 Input measurements

The LHCb collaboration has published three analyses relevant to this paper
based on the data corresponding to an integrated luminosity of
$1.0\mbox{\,fb}^{-1}$ using $pp$ collisions at a centre-of-mass energy of
$7\mathrm{\,Te\kern-1.00006ptV}$, recorded in 2011. They are a GGSZ
measurement of $B^{\pm}\\!\rightarrow DK^{\pm}$ decays, where the $D$ meson is
reconstructed in the $D\rightarrow K^{0}_{\rm\scriptscriptstyle
S}\pi^{+}\pi^{-}$ and $D\rightarrow K^{0}_{\rm\scriptscriptstyle S}K^{+}K^{-}$
final states [24]; a GLW/ADS measurement of $B^{\pm}\\!\rightarrow DK^{\pm}$
and $B^{\pm}\\!\rightarrow D\pi^{\pm}$ decays, where the $D$ meson is
reconstructed in charged two-body final states [25]; and an ADS measurement of
$B^{\pm}\\!\rightarrow DK^{\pm}$ and $B^{\pm}\\!\rightarrow D\pi^{\pm}$
decays, where the $D$ meson is reconstructed in charged four-body final states
[26]. In addition, inputs from a combination of experimental data performed by
the HFAG, to constrain the effect of direct $C\\!P$ violation in $D$ decays
[27], and measurements from the LHCb collaboration [18] and the CLEO
collaboration [13], to constrain the hadronic parameters of the $D$ system,
are included. Ref. [13] includes itself inputs by the HFAG.

###  2.1 Measurements from $B^{\pm}\rightarrow D[\rightarrow
K^{0}_{\rm\scriptscriptstyle S}h^{+}h^{-}]K^{\pm}$ decays

The GGSZ method [12] proposes the use of self-conjugate three-body $D$ decays
in the measurement of $\gamma$ from $B^{\pm}\\!\rightarrow DK^{\pm}$
processes. The variables $x_{\pm}$ and $y_{\pm}$, defined as

$\displaystyle x_{\pm}$
$\displaystyle=r_{B}^{K}\cos(\delta_{B}^{K}\pm\gamma)\,,$ (3) $\displaystyle
y_{\pm}$ $\displaystyle=r_{B}^{K}\sin(\delta_{B}^{K}\pm\gamma)\,,$ (4)

are obtained from a fit to the Dalitz plane of $D\rightarrow
K^{0}_{\rm\scriptscriptstyle S}\pi^{+}\pi^{-}$ and $D\rightarrow
K^{0}_{\rm\scriptscriptstyle S}K^{+}K^{-}$ decays, separately for $B^{+}$ and
$B^{-}$ decays. The measurement, performed by LHCb, is reported in Ref. [24].
The study makes no model-dependent assumption on the variation of the strong
phase of the $D\rightarrow K^{0}_{\rm\scriptscriptstyle S}h^{+}h^{-}$
amplitudes, but instead uses measurements of this quantity from CLEO [28], as
input. The reported results are

$\displaystyle x_{-}$ $\displaystyle=(\phantom{-1}0.0\pm 4.3\pm 1.5\pm
0.6)\times 10^{-2}\,,$ (5) $\displaystyle y_{-}$
$\displaystyle=(\phantom{-1}2.7\pm 5.2\pm 0.8\pm 2.3)\times 10^{-2}\,,$ (6)
$\displaystyle x_{+}$ $\displaystyle=(-10.3\pm 4.5\pm 1.8\pm 1.4)\times
10^{-2}\,,$ (7) $\displaystyle y_{+}$ $\displaystyle=(\phantom{1}{-}0.9\pm
3.7\pm 0.8\pm 3.0)\times 10^{-2}\,,$ (8)

where the first uncertainty is statistical, the second is systematic, and the
third is due to the external CLEO measurement. The non-vanishing statistical
correlations are $\rho(x_{-},y_{-})=-0.11$, $\rho(x_{+},y_{+})=+0.17$, and the
relevant systematic correlations are $\rho(x_{-},y_{-})=-0.05$, and
$\rho(x_{+},y_{+})=+0.36$.

The GGSZ method can also be applied to $B^{\pm}\\!\rightarrow D\pi^{\pm}$
final states. In Ref. [24] this was not performed, since these final states
were needed to control the efficiency variation across the Dalitz plot. The
effect of $D^{0}$–$\kern 1.99997pt\overline{\kern-1.99997ptD}{}^{0}$ mixing in
the measurement of the $x_{\pm}$ and $y_{\pm}$ in Eqns. 5–8 is suppressed,
leading to a negligible effect in the extraction of $\gamma$ [16, 17].

###  2.2 Measurements from $B^{\pm}\rightarrow D[\rightarrow
h^{+}h^{-}]h^{\pm}$ decays

The $D$ decay modes considered in the analysis of two-body $D$ final states
[25] are $D\rightarrow K^{+}K^{-}$, $D\rightarrow\pi^{+}\pi^{-}$, the favoured
decay $D\rightarrow K^{-}\pi^{+}$, where the kaon charge matches that of the
$h^{\pm}$ track from the $B^{\pm}\rightarrow Dh^{\pm}$ decay (called $K\pi$ in
the following), and the suppressed decay $D\rightarrow\pi^{-}K^{+}$, where the
kaon charge is opposite that of the $h^{\pm}$ track (called $\pi K$ in the
following). Building on the initial GLW/ADS ideas [8, 9, 10, 11], a set of 13
observables was defined by forming ratios of decay rates, defined below, such
that many systematic uncertainties cancel. The charge-averaged ratios of
$B^{\pm}\\!\rightarrow DK^{\pm}$ and $B^{\pm}\\!\rightarrow D\pi^{\pm}$ decays
are

$\displaystyle R_{K/\pi}^{f}$ $\displaystyle=\frac{\Gamma(B^{-}\rightarrow
D[\rightarrow f]K^{-})+\Gamma(B^{+}\rightarrow
D[\rightarrow\overline{f}]K^{+})}{\Gamma(B^{-}\rightarrow D[\rightarrow
f]\pi^{-})+\Gamma(B^{+}\rightarrow D[\rightarrow\overline{f}]\pi^{+})}\,,$ (9)

where $f$ is the relevant final state. The ratios $R_{K/\pi}^{f}$ are related
to $\gamma$ and the hadronic parameters through

$\displaystyle R_{K/\pi}^{f}$ $\displaystyle=R_{\rm
cab}\frac{1+(r_{B}^{K}r_{f})^{2}+2r_{B}^{K}r_{f}\kappa\cos(\delta_{B}^{K}-\delta_{f})\cos\gamma+M^{K}_{-}+M^{K}_{+}}{1+(r_{B}^{\pi}r_{f})^{2}+2r_{B}^{\pi}r_{f}\kappa\cos(\delta_{B}^{\pi}-\delta_{f})\cos\gamma+M^{\pi}_{-}+M^{\pi}_{+}}\,,$
(10)

for the favoured final state $f=K\pi$, where the coherence factor $\kappa$ in
Eq. 10 (and in all following equations in this Section) is unity for two-body
decays, and through

$\displaystyle R_{K/\pi}^{f}$ $\displaystyle=R_{\rm
cab}\frac{1+(r_{B}^{K})^{2}+2r_{B}^{K}\cos\delta_{B}^{K}\cos\gamma}{1+(r_{B}^{\pi})^{2}+2r_{B}^{\pi}\cos\delta_{B}^{\pi}\cos\gamma}\,,$
(11)

for $f=KK,\pi\pi$. The $D$ mixing correction terms $M^{h}_{\pm}$ are, at
leading order in $x_{D}$ and $y_{D}$, and neglecting $C\\!P$ violation in $D$
mixing, given by [14]

$\displaystyle M^{h}_{\pm}$ $\displaystyle=\left(\kappa
r_{f}((r_{B}^{h})^{2}-1)\sin\delta_{f}+r_{B}^{h}(1-r_{f}^{2})\sin(\delta_{B}^{h}\pm\gamma)\right)\,a_{D}\,x_{D}$
$\displaystyle-\left(\kappa
r_{f}((r_{B}^{h})^{2}+1)\cos\delta_{f}+r_{B}^{h}(1+r_{f}^{2})\cos(\delta_{B}^{h}\pm\gamma)\right)\,a_{D}\,y_{D}\,.$
(12)

The $D$ mixing corrections depend on the $D$ decay time acceptance and
resolution in the reconstruction of $B^{\pm}\\!\rightarrow Dh^{\pm}$ decays
[17]. The coefficient $a_{D}$ parameterises their effect. It takes the value
of $a_{D}=1$ in case of an ideal, flat acceptance and negligible time
resolution. For a realistic acceptance and resolution model present in the
GLW/ADS analysis of Ref. [25], it is estimated to be $a_{D}=1.20\pm 0.04$,
where the uncertainty can be safely neglected in this combination. For $C\\!P$
even final states of the $D$ meson, the mixing corrections cancel exactly in
Eq. 11 (and 15), as in this case $\kappa=1$, $r_{f}=1$, $\delta_{f}=0$. The
charge asymmetries are

$\displaystyle A_{h}^{f}$ $\displaystyle=\frac{\Gamma(B^{-}\rightarrow
D[\rightarrow f]h^{-})-\Gamma(B^{+}\rightarrow
D[\rightarrow\overline{f}]h^{+})}{\Gamma(B^{-}\rightarrow D[\rightarrow
f]h^{-})+\Gamma(B^{+}\rightarrow D[\rightarrow\overline{f}]h^{+})}\,,$ (13)

which are related to $\gamma$ and the hadronic parameters through

$\displaystyle A_{h}^{f}$
$\displaystyle=\frac{2r_{B}^{h}r_{f}\kappa\sin(\delta_{B}^{h}-\delta_{f})\sin\gamma+M^{h}_{-}-M^{h}_{+}}{1+(r_{B}^{h}r_{f})^{2}+2r_{B}^{h}r_{f}\kappa\cos(\delta_{B}^{h}-\delta_{f})\cos\gamma+M^{h}_{-}+M^{h}_{+}}\,,$
(14)

for the favoured final state $f=K\pi$, and through

$\displaystyle A_{h}^{f}$
$\displaystyle=\frac{2r_{B}^{h}\sin\delta_{B}^{h}\sin\gamma}{1+(r_{B}^{h})^{2}+2r_{B}^{h}\cos\delta_{B}^{h}\cos\gamma}\,,$
(15)

for $f=KK$, $\pi\pi$, where $r_{B}^{h}$ denotes $r_{B}^{K}$ and $r_{B}^{\pi}$.
Finally, the non charge-averaged ratios of suppressed and favoured $D$ final
states are

$\displaystyle R_{h}^{\pm}$ $\displaystyle=\frac{\Gamma(B^{\pm}\rightarrow
D[\rightarrow f_{\rm sup}]h^{\pm})}{\Gamma(B^{\pm}\rightarrow D[\rightarrow
f]h^{\pm})}$
$\displaystyle=\frac{r_{f}^{2}+(r_{B}^{h})^{2}+2r_{B}^{h}r_{f}\kappa\cos(\delta_{B}^{h}+\delta_{f}\pm\gamma)-[M^{h}_{\pm}]_{\rm
sup}}{1+(r_{B}^{h}r_{f})^{2}+2r_{B}^{h}r_{f}\kappa\cos(\delta_{B}^{h}-\delta_{f}\pm\gamma)+M^{h}_{\pm}}\,,$
(16)

where $f_{\rm sup}=\pi K$ is the suppressed final state, and $f=K\pi$ the
allowed one. The suppressed $D$ mixing correction terms are given, at leading
order in $x_{D}$ and $y_{D}$, by

$\displaystyle[M^{h}_{\pm}]_{\rm sup}$ $\displaystyle=\left(\kappa
r_{f}((r_{B}^{h})^{2}-1)\sin\delta_{f}+r_{B}^{h}(1-r_{f}^{2})\sin(\delta_{B}^{h}\pm\gamma)\right)\,a_{D}\,x_{D}$
$\displaystyle+\left(\kappa
r_{f}((r_{B}^{h})^{2}+1)\cos\delta_{f}+r_{B}^{h}(1+r_{f}^{2})\cos(\delta_{B}^{h}\pm\gamma)\right)\,a_{D}\,y_{D}\,.$
(17)

The combination makes use of all $\gamma$-sensitive observables determined in
the GLW/ADS analysis. The full set, taken from the two-body analysis [25], is

$\displaystyle R_{K/\pi}^{K\pi}$
$\displaystyle=\phantom{-}0.0774\phantom{0}\pm 0.0012\phantom{0}\pm 0.0018\,,$
$\displaystyle R_{K/\pi}^{KK}$ $\displaystyle=\phantom{-}0.0773\phantom{0}\pm
0.0030\phantom{0}\pm 0.0018\,,$ $\displaystyle R_{K/\pi}^{\pi\pi}$
$\displaystyle=\phantom{-}0.0803\phantom{0}\pm 0.0056\phantom{0}\pm 0.0017\,,$
$\displaystyle A_{\pi}^{K\pi}$ $\displaystyle=-0.0001\phantom{0}\pm
0.0036\phantom{0}\pm 0.0095\,,$ $\displaystyle A_{K}^{K\pi}$
$\displaystyle=\phantom{-}0.0044\phantom{0}\pm 0.0144\phantom{0}\pm 0.0174\,,$
$\displaystyle A_{K}^{KK}$ $\displaystyle=\phantom{-}0.148\phantom{00}\pm
0.037\phantom{00}\pm 0.010\,,$ $\displaystyle A_{K}^{\pi\pi}$
$\displaystyle=\phantom{-}0.135\phantom{00}\pm 0.066\phantom{00}\pm 0.010\,,$
$\displaystyle A_{\pi}^{KK}$ $\displaystyle=-0.020\phantom{00}\pm
0.009\phantom{00}\pm 0.012\,,$ $\displaystyle A_{\pi}^{\pi\pi}$
$\displaystyle=-0.001\phantom{00}\pm 0.017\phantom{00}\pm 0.010\,,$
$\displaystyle R_{K}^{-}$ $\displaystyle=\phantom{-}0.0073\phantom{0}\pm
0.0023\phantom{0}\pm 0.0004\,,$ $\displaystyle R_{K}^{+}$
$\displaystyle=\phantom{-}0.0232\phantom{0}\pm 0.0034\phantom{0}\pm 0.0007\,,$
$\displaystyle R_{\pi}^{-}$ $\displaystyle=\phantom{-}0.00469\pm 0.00038\pm
0.00008\,,$ $\displaystyle R_{\pi}^{+}$ $\displaystyle=\phantom{-}0.00352\pm
0.00033\pm 0.00007\,,$

where the first uncertainty is statistical and the second systematic. Their
statistical correlations, not previously published, are given in Table 2.

Table 2: Statistical correlations of the $B^{\pm}\rightarrow Dh^{\pm}$, $D\rightarrow hh$ analysis [25]. | $A_{K}^{KK}$ | $A_{\pi}^{KK}$ | $A_{K}^{\pi\pi}$ | $A_{\pi}^{\pi\pi}$ | $A_{K}^{K\pi}$ | $A_{\pi}^{K\pi}$ | $R_{K/\pi}^{\pi\pi}$ | $R_{K/\pi}^{KK}$ | $R_{K/\pi}^{K\pi}$ | $R_{K}^{-}$ | $R_{\pi}^{-}$ | $R_{K}^{+}$ | $R_{\pi}^{+}$
---|---|---|---|---|---|---|---|---|---|---|---|---|---
$A_{K}^{KK}$ | 1 | -0.029 | 0 | 0 | 0 | 0 | -0.002 | -0.034 | -0.010 | -0.001 | 0 | 0 | 0
$A_{\pi}^{KK}$ |  | 1 | 0 | 0 | 0 | 0 | 0 | -0.003 | 0 | 0 | 0 | 0 | 0
$A_{K}^{\pi\pi}$ |  |  | 1 | -0.032 | 0 | 0 | -0.032 | -0.002 | -0.004 | -0.001 | 0 | 0 | 0
$A_{\pi}^{\pi\pi}$ |  |  |  | 1 | 0 | 0 | -0.004 | 0 | 0 | 0 | 0 | 0 | 0
$A_{K}^{K\pi}$ |  |  |  |  | 1 | -0.045 | 0 | 0 | 0.003 | 0.004 | 0 | -0.004 | -0.001
$A_{\pi}^{K\pi}$ |  |  |  |  |  | 1 | 0 | 0 | -0.001 | 0.004 | 0.002 | -0.004 | -0.002
$R_{K/\pi}^{\pi\pi}$ |  |  |  |  |  |  | 1 | 0.013 | 0.029 | 0.003 | 0.003 | 0.001 | 0.003
$R_{K/\pi}^{KK}$ |  |  |  |  |  |  |  | 1 | 0.053 | 0.005 | 0.005 | 0.002 | 0.004
$R_{K/\pi}^{K\pi}$ |  |  |  |  |  |  |  |  | 1 | -0.038 | 0.016 | -0.093 | 0.014
$R_{K}^{-}$ |  |  |  |  |  |  |  |  |  | 1 | -0.023 | 0.012 | 0.006
$R_{\pi}^{-}$ |  |  |  |  |  |  |  |  |  |  | 1 | 0.005 | 0.008
$R_{K}^{+}$ |  |  |  |  |  |  |  |  |  |  |  | 1 | -0.036
$R_{\pi}^{+}$ |  |  |  |  |  |  |  |  |  |  |  |  | 1

Direct $C\\!P$ asymmetries in $D^{0}\rightarrow K^{+}K^{-}$ and
$D^{0}\rightarrow\pi^{+}\pi^{-}$ decays have been measured [19, 20]. While the
effect on the charge averaged ratios $R_{K/\pi}^{KK}$ and $R_{K/\pi}^{\pi\pi}$
is negligible [22], the observables $A_{h}^{KK}$ and $A_{h}^{\pi\pi}$ are
modified by adding the respective direct $C\\!P$ asymmetry $A_{C\\!P}^{\rm
dir}$ to the right-hand side of Eq. 15. This is valid up to neglecting a small
weak phase in the $D$ decay [22]. The HFAG results on $A_{C\\!P}^{\rm dir}$
[27] are included in this combination

$\displaystyle A_{C\\!P}^{\rm dir}(KK)$ $\displaystyle=(-0.31\pm 0.24)\times
10^{-2}\,,$ $\displaystyle A_{C\\!P}^{\rm dir}(\pi\pi)$
$\displaystyle=(+0.36\pm 0.25)\times 10^{-2}\,.$

These quantities are correlated, $\rho(A_{C\\!P}^{\rm dir}(KK),A_{C\\!P}^{\rm
dir}(\pi\pi))=+0.80$, and therefore they are constrained to their observed
values by means of a two-dimensional correlated Gaussian PDF. The inclusion of
the result on $A_{C\\!P}^{\rm dir}(KK)-A_{C\\!P}^{\rm dir}(\pi\pi)$ [19],
which is statistically independent from the HFAG average, is found to have no
effect on the combination.

###  2.3 Measurements from $B^{\pm}\rightarrow D[\rightarrow
K^{\pm}\pi^{\mp}\pi^{+}\pi^{-}]h^{\pm}$ decays

The $D$ four-body decay modes considered in the analysis of Ref. [26] are the
favoured $D\rightarrow K^{-}\pi^{+}\pi^{-}\pi^{+}$, and the suppressed
$D\rightarrow\pi^{-}K^{+}\pi^{-}\pi^{+}$ final states. In a similar manner to
the two-body GLW/ADS analysis, seven observables are defined as ratios of
decay rates. Their relations to $\gamma$ and the hadronic parameters are fully
analogous and given by Eqs. 10, 14, and 16, with $f=K\pi\pi\pi$ and $f_{\rm
sup}=\pi K\pi\pi$. The $C\\!P$-violating effects are diluted due to the $D$
decay proceeding through a range of resonances that can only interfere in
limited regions of the four-body phase space. This dilution is accounted for
by multiplying each interference term by a coherence factor
$\kappa=\kappa_{K3\pi}$. The $D$ decay time acceptance and resolution model is
identical to that present in the two-body GLW/ADS analysis of Ref. [25]. The
seven observables, taken from the four-body analysis reported in Ref. [26],
are

$\displaystyle R_{K/\pi}^{K3\pi}$
$\displaystyle=\phantom{-}0.0765\phantom{0}\pm 0.0017\phantom{0}\pm 0.0026\,,$
$\displaystyle A_{\pi}^{K3\pi}$ $\displaystyle=-0.006\phantom{00}\pm
0.005\phantom{00}\pm 0.010\,,$ $\displaystyle A_{K}^{K3\pi}$
$\displaystyle=-0.026\phantom{00}\pm 0.020\phantom{00}\pm 0.018\,,$
$\displaystyle R_{K-}^{K3\pi}$ $\displaystyle=\phantom{-}0.0071\phantom{0}\pm
0.0034\phantom{0}\pm 0.0008\,,$ $\displaystyle R_{K+}^{K3\pi}$
$\displaystyle=\phantom{-}0.0155\phantom{0}\pm 0.0042\phantom{0}\pm 0.0010\,,$
$\displaystyle R_{\pi-}^{K3\pi}$ $\displaystyle=\phantom{-}0.00400\pm
0.00052\pm 0.00011\,,$ $\displaystyle R_{\pi+}^{K3\pi}$
$\displaystyle=\phantom{-}0.00316\pm 0.00046\pm 0.00011\,,$

where the first uncertainty is statistical and the second systematic. The
statistical correlations between these variables, not previously published,
are presented in Table 3.

Table 3: Statistical correlations of the $B^{\pm}\rightarrow Dh^{\pm}$, $D\rightarrow K\pi\pi\pi$ analysis [26]. | $R_{K/\pi}^{K3\pi}$ | $A_{K}^{K3\pi}$ | $A_{\pi}^{K3\pi}$ | $R_{K-}^{K3\pi}$ | $R_{K+}^{K3\pi}$ | $R_{\pi-}^{K3\pi}$ | $R_{\pi+}^{K3\pi}$
---|---|---|---|---|---|---|---
$R_{K/\pi}^{K3\pi}$ | 1 | 0.003 | 0.001 | -0.060 | -0.024 | 0.017 | 0.021
$A_{K}^{K3\pi}$ |  | 1 | -0.035 | -0.007 | 0.006 | -0.002 | 0.002
$A_{\pi}^{K3\pi}$ |  |  | 1 | -0.006 | 0.008 | -0.002 | 0.005
$R_{K-}^{K3\pi}$ |  |  |  | 1 | 0.043 | 0.006 | 0.029
$R_{K+}^{K3\pi}$ |  |  |  |  | 1 | 0.022 | 0.025
$R_{\pi-}^{K3\pi}$ |  |  |  |  |  | 1 | 0.032
$R_{\pi+}^{K3\pi}$ |  |  |  |  |  |  | 1

###  2.4 Measurement of the hadronic parameters of the $D$ system from
$D^{0}\rightarrow K^{\pm}\pi^{\mp}$, $K^{\pm}\pi^{\mp}\pi^{+}\pi^{-}$ decays
by CLEO

The two- and four-body ADS measurements both reach their best sensitivity when
combined with knowledge of the hadronic parameters of the $D$ decay. These
are, for the $D^{0}\rightarrow K^{\pm}\pi^{\mp}$ decays, the amplitude ratio
$r_{K\pi}$ and the strong phase difference $\delta_{K\pi}$. The hadronic
parameters of the $D^{0}\rightarrow K^{\pm}\pi^{\mp}\pi^{+}\pi^{-}$ decays are
the ratio $r_{K3\pi}$, the phase $\delta_{K3\pi}$ and the coherence
factor111Note that Ref. [13] uses the symbol $R_{K3\pi}$ to denote the
coherence factor. $\kappa_{K3\pi}$. All of these parameters are constrained by
a CLEO measurement [13], where a combined fit is performed, which includes
information on the $D$ mixing parameters and the Cabibbo-favoured branching
fractions of the $D$ decay through the following relationship

$\displaystyle\frac{\Gamma(D^{0}\rightarrow f_{\rm
sup})}{\Gamma(D^{0}\rightarrow f_{\rm
fav})}=r_{f}^{2}\left[1-\frac{y_{D}}{r_{f}}\kappa\cos\delta_{f}+\frac{x_{D}}{r_{f}}\kappa\sin\delta_{f}+\frac{x_{D}^{2}+y_{D}^{2}}{2r_{f}^{2}}\right]\,,$
(18)

where $r_{f}=r_{K\pi}$ $(r_{K3\pi})$, $\delta_{f}=\delta_{K\pi}$
$(\delta_{K3\pi})$, and $\kappa=1$ $(\kappa_{K3\pi})$, for $D^{0}\rightarrow
K^{\pm}\pi^{\mp}$ ($K^{\pm}\pi^{\mp}\pi^{+}\pi^{-}$) decays. All of these
parameters are included in the combination, although the dependence of
$\gamma$ on the $D$ mixing parameters and the Cabibbo-favoured branching
fractions is small compared to the current statistical precision. The central
values and the uncertainties given in Table 4 are reproduced from the analysis
by the CLEO collaboration reported in Ref. [13]. The covariance matrix (see
Table VI in Ref. [13]) is also used, though it is not reproduced here. The
parameters $(\delta_{K3\pi},\kappa_{K3\pi})$ exhibit a non-Gaussian two-
dimensional likelihood (see Fig. 2b in Ref. [13]), and this likelihood is used
in the combination [29]. Their central values and profile-likelihood
uncertainties are $\kappa_{K3\pi}=0.33^{+0.26}_{-0.23}$ and
$\delta_{K3\pi}=(114^{+26}_{-23})^{\circ}$. Correlations of $\delta_{K3\pi}$
and $\kappa_{K3\pi}$ to other parameters are neglected.

Table 4: Results of the CLEO measurement [13]. Observable | Central value and uncertainty
---|---
$\delta_{K\pi}$ | $(-151.5^{+9.6}_{-9.5})^{\circ}$
$x_{D}$ | $(0.96\pm 0.25)\times 10^{-2}$
$y_{D}$ | $(0.81\pm 0.16)\times 10^{-2}$
${\cal B}(D^{0}\rightarrow K^{-}\pi^{+})$ | $(3.89\pm 0.05)\times 10^{-2}$
${\cal B}(D^{0}\rightarrow\pi^{-}K^{+})$ | $(1.47\pm 0.07)\times 10^{-4}$
${\cal B}(D^{0}\rightarrow K^{-}\pi^{+}\pi^{-}\pi^{+})$ | $(7.96\pm 0.19)\times 10^{-2}$
${\cal B}(D^{0}\rightarrow\pi^{-}K^{+}\pi^{-}\pi^{+})$ | $(2.65\pm 0.19)\times 10^{-4}$

###  2.5 Measurement from $D^{0}\rightarrow K^{\pm}\pi^{\mp}$ decays by LHCb

The $D$ mixing parameters $x_{D}$ and $y_{D}$ are constrained in addition by
an LHCb measurement of $D^{0}\rightarrow K^{\pm}\pi^{\mp}$ decays [18]. Three
observables are defined, $R_{D}$, $y^{\prime}_{D}$, and $x_{D}^{\prime 2}$,
that are related to the $D$ system parameters through the following
relationships

$\displaystyle R_{D}$ $\displaystyle=r_{K\pi}^{2}\,,$ (19) $\displaystyle
y^{\prime}_{D}$
$\displaystyle=x_{D}\sin\delta_{K\pi}-y_{D}\cos\delta_{K\pi}\,,$ (20)
$\displaystyle x_{D}^{\prime 2}$
$\displaystyle=\left(x_{D}\cos\delta_{K\pi}+y_{D}\sin\delta_{K\pi}\right)^{2}\,,$
(21)

where a phase shift of $180^{\circ}$ was introduced to $\delta_{K\pi}$ to be
in accordance with the phase convention adopted in this Letter. In Ref. [18],
the measured central values of the observables are $R_{D}=(3.52\pm 0.15)\times
10^{-3}$, $y^{\prime}_{D}=(7.2\pm 2.4)\times 10^{-3}$, and $x_{D}^{\prime
2}=(-0.09\pm 0.13)\times 10^{-3}$, where the error includes both statistical
and systematic uncertainties. These observables are strongly correlated,
$\rho(R_{D},y^{\prime}_{D})=-0.95$, $\rho(y^{\prime}_{D},x_{D}^{\prime
2})=-0.97$, and $\rho(x_{D}^{\prime 2},R_{D})=+0.88$. They are included by
means of a three-dimensional correlated Gaussian PDF.

##  3 Statistical interpretation

The evaluation of this combination follows a frequentist approach. A
$\chi^{2}$-function is defined as
$\chi^{2}(\vec{\alpha})=-2\ln\mathcal{L}(\vec{\alpha})$, where
$\mathcal{L}(\vec{\alpha})$ is defined in Eq. 1. The best-fit point is given
by the global minimum of the $\chi^{2}$-function,
$\chi^{2}(\vec{\alpha}_{\min})$. To evaluate the confidence level for a given
value of a certain parameter, say $\gamma=\gamma_{0}$ in the following, the
value of the $\chi^{2}$-function at the new minimum is considered,
$\chi^{2}(\vec{\alpha}^{\prime}_{\min}(\gamma_{0}))$. This also defines the
profile likelihood function
$\hat{\mathcal{L}}(\gamma_{0})=\exp(-\chi^{2}(\vec{\alpha}^{\prime}_{\min})/2)$.
Then a test statistic is defined as
$\Delta\chi^{2}=\chi^{2}(\vec{\alpha}^{\prime}_{\min})-\chi^{2}(\vec{\alpha}_{\min})$.
The $p$-value, or $1-{\rm CL}$, is calculated by means of a Monte Carlo
procedure, described in Ref. [30] and briefly recapitulated here. For each
value of $\gamma_{0}$:

  1. 1.

$\Delta\chi^{2}$ is calculated;

  2. 2.

a set of pseudoexperiments $\vec{A}_{j}$ is generated using Eq. 1 with
parameters $\vec{\alpha}$ set to $\vec{\alpha}^{\prime}_{\min}$ as the PDF;

  3. 3.

$\Delta\chi^{2\prime}$ of the pseudoexperiment is calculated by replacing
$\vec{A}_{\rm obs}\rightarrow\vec{A}_{j}$ and minimising with respect to
$\vec{\alpha}$, once with $\gamma$ as a free parameter, and once with $\gamma$
fixed to $\gamma_{0}$;

  4. 4.

$1-{\rm CL}$ is calculated as the fraction of pseudoexperiments which perform
worse ($\Delta\chi^{2}<\Delta\chi^{2\prime}$) than the measured data.

This method is sometimes known as the “$\hat{\mu}$”, or the “plug-in” method.
Its coverage cannot be guaranteed [30] for the full parameter space, but is
verified for the best-fit point. The reason is, that at each point
$\gamma_{0}$, the nuisance parameters, i.e. the components of $\vec{\alpha}$
other than the parameter of interest, are set to their best-fit values for
this point, as opposed to computing an $n$-dimensional confidence belt, which
is computationally very demanding.

In case of the CLEO likelihood for $\kappa_{K3\pi}$ and $\delta_{K3\pi}$, it
is assumed that the true PDF, for any assumed true value of $\kappa_{K3\pi}$
and $\delta_{K3\pi}$, can be described by a shifted version of the likelihood
profile. In the non-physical range, $\kappa_{K3\pi}\notin[0,1]$, the
likelihood profile is not available. It is extrapolated into the non-physical
range using Gaussian tails that correspond to the published uncertainties of
the central value. If $H(x,y)$ denotes the provided likelihood profile, with a
maximum at position $(\hat{x},\hat{y})$, it is transformed as $f_{i}(x_{\rm
obs},y_{\rm obs}|x,y)\propto H_{i}(x-x_{\rm obs}+\hat{x},~{}y-y_{\rm
obs}+\hat{y})$, with the abbreviation $(x,y)=(\kappa_{K3\pi},\delta_{K3\pi})$.

##  4 Results

Three different combinations are presented. First, only the parts
corresponding to $B^{\pm}\\!\rightarrow DK^{\pm}$ decays of the two- and four-
body GLW/ADS measurements [25, 26] are combined with the GGSZ [24]
measurement. Then, only the $B^{\pm}\\!\rightarrow D\pi^{\pm}$ parts of the
two- and four-body GLW/ADS measurements are combined. Finally, the full
$B^{\pm}\\!\rightarrow Dh^{\pm}$ combination is computed. It is difficult to
disentangle the $B^{\pm}\\!\rightarrow DK^{\pm}$ and $B^{\pm}\\!\rightarrow
D\pi^{\pm}$ measurements, because the observed ratios of Eq. 9 necessarily
contain information on both systems. These ratios are therefore included in
the $B^{\pm}\\!\rightarrow DK^{\pm}$ combination, but not in the
$B^{\pm}\\!\rightarrow D\pi^{\pm}$ combination. To include them in the
$B^{\pm}\\!\rightarrow DK^{\pm}$ combination, the denominator in the second
term of Eq. 10 is assumed to equal unity, neglecting a correction smaller than
$0.04$, such that effects of hadronic parameters in the $B^{\pm}\\!\rightarrow
D\pi^{\pm}$ system are avoided. The separate $DK^{\pm}$ ($D\pi^{\pm}$)
combination contains 29 (22) observables, and the full combination contains 38
observables, as 13 observables from CLEO, HFAG, and Ref. [18] are common to
both separate combinations. The results are summarised in Tables 5–7, and
illustrated in Figs. 1–3. The equations of Sect. 2 are invariant under the
simultaneous transformation $\gamma\rightarrow\gamma+180^{\circ}$,
$\delta\rightarrow\delta+180^{\circ}$, where
$\delta=\delta_{B}^{K},\delta_{B}^{\pi}$. All results on $\gamma$,
$\delta_{B}^{K}$, and $\delta_{B}^{\pi}$ are expressed modulo $180^{\circ}$,
and only the solution most consistent with the average computed by CKMfitter
and UTfit is shown. Figure 4 shows two-dimensional profile likelihood contours
of the full combination, where the discrete symmetry is apparent in subfigures
(b) and (d). The $DK^{\pm}$ combination results in confidence intervals for
$\gamma$ that are symmetric and almost Gaussian up to $95\%$ CL. Beyond that a
secondary, local minimum of $\chi^{2}(\vec{\alpha}^{\prime}_{\rm min})$ causes
a much enlarged interval at $99\%$ CL. The $D\pi^{\pm}$ combination results in
unexpectedly small confidence intervals at $68\%$ CL. This can be explained by
an upward fluctuation of $r_{B}^{\pi}$, since again the uncertainty of
$\gamma$ scales roughly like $1/r_{B}^{\pi}$. The ratio $r_{B}^{\pi}$ is
expected to be
$r_{B}^{\pi}\approx|(V_{ub}^{*}V_{cd}^{\phantom{*}})/(V_{cb}^{*}V_{ud}^{\phantom{*}})|\times|C|/|T+C|\approx
0.006$, where $C$ and $T$ describe the magnitudes of the colour-suppressed and
tree amplitudes governing $B^{\pm}\\!\rightarrow D\pi^{\pm}$ decays, with
their numerical values estimated from Ref. [31]. Within the $95\%$ CL
interval, $r_{B}^{\pi}$ is well consistent with this expectation, and no
constraints on $\gamma$ are set. The high value of $r_{B}^{\pi}$ also affects
the full combination.

Table 5: Confidence intervals and best-fit values of the $DK^{\pm}$ combination for $\gamma$, $\delta_{B}^{K}$, and $r_{B}^{K}$. Quantity |  $DK^{\pm}$ combination
---|---
$\gamma$ | $72.0^{\circ}$
68% CL | $[56.4,86.7]^{\circ}$
95% CL | $[42.6,99.6]^{\circ}$
$\delta_{B}^{K}$ | $112^{\circ}$
68% CL | $[96,126]^{\circ}$
95% CL | $[80,136]^{\circ}$
$r_{B}^{K}$ | $0.089$
68% CL | $[0.080,0.098]$
95% CL | $[0.071,0.107]$
Table 6: Confidence intervals and best-fit values for the $D\pi^{\pm}$ combination for $\gamma$, $\delta_{B}^{\pi}$, and $r_{B}^{\pi}$. The corrections to the $\gamma$ intervals for undercoverage and neglected systematic correlations, as described in Sect. 5, are not yet applied. Quantity |  $D\pi^{\pm}$ combination
---|---
$\gamma$ | $18.9^{\circ}$
68% CL | $[8.9,80.2]^{\circ}\,\cup\,[169.1,175.7]^{\circ}$
95% CL | no constraint
$\delta_{B}^{\pi}$ | $261^{\circ}$
68% CL | $[213,229]^{\circ}\,\cup\,[249,331]^{\circ}$
95% CL | no constraint
$r_{B}^{\pi}$ | $0.015$
68% CL | $[0.006,0.056]$
95% CL | $[0.001,0.073]$
Table 7: Confidence intervals and best-fit values for the $DK^{\pm}$ and $D\pi^{\pm}$ combination for $\gamma$, $r_{B}^{K}$, $\delta_{B}^{K}$, $r_{B}^{\pi}$, and $\delta_{B}^{\pi}$. The corrections to the $\gamma$ intervals for undercoverage and neglected systematic correlations, as described in Sect. 5, are not yet applied. Quantity |  $DK^{\pm}$ and $D\pi^{\pm}$ combination
---|---
$\gamma$ | $72.6^{\circ}$
68% CL | $[56.7,81.7]^{\circ}$
95% CL | $[41.2,92.3]^{\circ}$
$r_{B}^{K}$ | $0.089$
68% CL | $[0.080,0.097]$
95% CL | $[0.071,0.105]$
$\delta_{B}^{K}$ | $112^{\circ}$
68% CL | $[96,125]^{\circ}$
95% CL | $[79,136]^{\circ}$
$r_{B}^{\pi}$ | $0.015$
68% CL | $[0.006,0.027]$
95% CL | $[0.002,0.036]$
$\delta_{B}^{\pi}$ | $315^{\circ}$
68% CL | $[269,332]$
95% CL | no constraint

\begin{overpic}[width=223.0721pt]{figs/gammacombo_dkonlyDmixeqDmixmeas_d_dk_pluginOnly_log.pdf}\put(190.0,135.0){a)}\end{overpic}\begin{overpic}[width=223.0721pt]{figs/gammacombo_dkonlyDmixeqDmixmeas_r_dk_pluginOnly_log.pdf}\put(190.0,135.0){b)}\end{overpic}

\begin{overpic}[width=223.0721pt]{figs/gammacombo_dkonlyDmixeqDmixmeas_g_pluginOnly_log.pdf}
\put(190.0,135.0){c)}\end{overpic}

Figure 1: Graphs showing $1-{\rm CL}$ for (a) $\delta_{B}^{K}$, (b)
$r_{B}^{K}$, and (c) $\gamma$, for the $DK^{\pm}$ combination of the two- and
four-body GLW/ADS and the $DK^{\pm}$ GGSZ measurements. The reported numbers
correspond to the best-fit values and the uncertainties are computed using the
respective $68.3\%$ CL confidence interval shown in Table 5.

\begin{overpic}[width=223.0721pt]{figs/gammacombo_dpionlyDmixeqDmixmeas_d_dpi_pluginOnly_log.pdf}\put(190.0,135.0){a)}\end{overpic}\begin{overpic}[width=223.0721pt]{figs/gammacombo_dpionlyDmixeqDmixmeas_r_dpi_pluginOnly_log.pdf}\put(190.0,135.0){b)}\end{overpic}

\begin{overpic}[width=223.0721pt]{figs/gammacombo_dpionlyDmixeqDmixmeas_g_pluginOnly_log.pdf}
\put(190.0,135.0){c)}\end{overpic}

Figure 2: Graphs showing $1-{\rm CL}$ for (a) $\delta_{B}^{\pi}$, (b)
$r_{B}^{\pi}$, and (c) $\gamma$, for the $D\pi^{\pm}$ combination of the two-
and four-body GLW/ADS measurements. The reported numbers correspond to the
best-fit values and the uncertainties are computed using appropriate $68.3\%$
CL confidence intervals shown in Table 6.

\begin{overpic}[width=223.0721pt]{figs/gammacombo_dkdpiDmixeqDmixmeas_d_dk_pluginOnly_log.pdf}
\put(190.0,135.0){a)}\end{overpic}\begin{overpic}[width=223.0721pt]{figs/gammacombo_dkdpiDmixeqDmixmeas_d_dpi_pluginOnly_log.pdf}\put(190.0,135.0){b)}\end{overpic}

\begin{overpic}[width=223.0721pt]{figs/gammacombo_dkdpiDmixeqDmixmeas_r_dk_pluginOnly_log.pdf}
\put(190.0,135.0){c)}\end{overpic}\begin{overpic}[width=223.0721pt]{figs/gammacombo_dkdpiDmixeqDmixmeas_r_dpi_pluginOnly_log.pdf}\put(190.0,135.0){d)}\end{overpic}

\begin{overpic}[width=223.0721pt]{figs/gammacombo_dkdpiDmixeqDmixmeas_g_pluginOnly_log.pdf}
\put(190.0,135.0){e)}\end{overpic}

Figure 3: Graphs showing $1-{\rm CL}$ for (a) $\delta_{B}^{K}$, (b)
$\delta_{B}^{\pi}$, (c) $r_{B}^{K}$, (d) $r_{B}^{\pi}$, and (e) $\gamma$, for
the full $DK^{\pm}$ and $D\pi^{\pm}$ combination. The reported numbers
correspond to the best-fit values and the uncertainties are computed using
appropriate $68.3\%$ CL confidence intervals shown in Table 7.

\begin{overpic}[width=223.0721pt]{figs/gammacombo_dkdpiDmixeqDmixmeas_g_r_dk.pdf}
\put(190.0,40.0){a)}\end{overpic}\begin{overpic}[width=223.0721pt]{figs/gammacombo_dkdpiDmixeqDmixmeas_g_d_dk.pdf}
\put(190.0,40.0){b)}\end{overpic}

\begin{overpic}[width=223.0721pt]{figs/gammacombo_dkdpiDmixeqDmixmeas_g_r_dpi.pdf}\put(190.0,40.0){c)}\end{overpic}\begin{overpic}[width=223.0721pt]{figs/gammacombo_dkdpiDmixeqDmixmeas_g_d_dpi.pdf}\put(190.0,40.0){d)}\end{overpic}

Figure 4: Profile likelihood contours of (a) $\gamma$ vs. $r_{B}^{K}$, (b)
$\gamma$ vs. $\delta_{B}^{K}$, (c) $\gamma$ vs. $r_{B}^{\pi}$, and (d)
$\gamma$ vs. $\delta_{B}^{\pi}$, for the full $DK^{\pm}$ and $D\pi^{\pm}$
combination. The contours are the $n\sigma$ profile likelihood contours, where
$\Delta\chi^{2}=n^{2}$ with $n=1,2$. The markers denote the best-fit values.
Subfigures (b) and (d) show the full angular range to visualize the symmetry,
while subfigures (a) and (c) are expressed modulo $180^{\circ}$.

##  5 Validation of results and systematic uncertainties

To assess the agreement between the various input measurements, the
probability $P$, that the observed dataset agrees better with the best-fit
model than a dataset generated assuming that model, is considered. It is
computed in two different ways. A first estimation of $P$ is obtained as the
$p$-value of a $\chi^{2}$ test on the value $\chi^{2}(\vec{\alpha}_{\rm
min})$, assuming it follows the $\chi^{2}$ distribution with a number of
degrees of freedom given by the difference of the the number of observables
$n_{\rm obs}$ and the number of fit parameters $n_{\rm fit}$. A more accurate
approach is to generate pseudodatasets $j$ at the best-fit value, and fit
these datasets with all parameters free. Then $P$ is given as the fraction of
pseudoexperiments that satisfy $(\chi^{2}_{j}>\chi^{2}_{\rm data})$. For this
test, the pseudoexperiments used for the plug-in method are re-evaluated. The
fit probability based on the $\chi^{2}$ distribution is well consistent with
that based on the pseudoexperiments, as shown in Table 8.

The statistical coverage of the plug-in method is not guaranteed. Therefore
the coverage is computed at the best-fit point for each of the three
combinations. This is done by generating pseudodatasets at the best-fit point,
and then, for each dataset, computing the $p$-value of the best-fit point
using the plug-in method. The coverage is then defined as the fraction
$\alpha$ in which the best-fit value of $\gamma$ has a larger $p$-value than
$\eta=68.27\%$, $\eta=95.45\%$, and $\eta=99.73\%$, for 1-, 2-, $3\sigma$,
respectively. The plug-in method slightly undercovers in the $D\pi^{\pm}$ and
full combinations, as shown in Table 9. The DK combination has exact coverage.
The same table also contains the coverage of the simpler interval setting
approach, in which the confidence intervals are defined by
$\Delta\chi^{2}=n^{2}$, where $n=1,2,3$. The profile likelihood approach was
found to significantly undercover. The $DK^{\pm}$ combination has exact
coverage. For the $D\pi^{\pm}$ and full combinations, the final plug-in
confidence intervals (Tabs. 6, 7) are scaled up by factors $\eta/\alpha$,
taken from Table 9.

In addition the confidence intervals were cross-checked using a method
inspired by Berger and Boos [32]. Instead of setting the nuisance parameters
$\vec{\theta}$ to their best-fit values when computing the $p$-value,
$p(\gamma_{0},\theta)$, $n_{\rm BB}=50$ alternative points are chosen, drawn
from an $(n_{\rm fit}-1)$-dimensional uniform distribution over a restricted
region $C_{\beta}$. Then, the $p$-value is given as $p_{\rm
BB}=\max_{\vec{\theta}\in C_{\beta}}p(\gamma_{0},\vec{\theta})+\beta$. Here,
$\beta$ is the probability that $\vec{\theta}$ lies outside $C_{\beta}$, and
$C_{\beta}$ is chosen large enough such that $\beta<10^{-4}$. This method is
more conservative than the nominal plug-in method, but is guaranteed to not
undercover for $n_{\rm BB}\rightarrow\infty$. The resulting intervals are only
slightly larger than the nominal ones.

Table 8: Numbers of observables $n_{\rm obs}$, numbers of free parameters in the fit $n_{\rm fit}$, the minimum $\chi^{2}$ at the best-fit point, and fit probabilities of the best-fit point for the three combinations. The quoted uncertainties are due to the limited number of pseudoexperiments. Combination | $n_{\rm obs}$ | $n_{\rm fit}$ | $\chi^{2}_{\rm min}$ |  $P$[%] ($\chi^{2}$ distribution) |  $P$[%] (pseudoexperiments)
---|---|---|---|---|---
$DK^{\pm}$ | 29 | 15 | $10.48$ | $72.6$ | $73.9\pm 0.2$
$D\pi^{\pm}$ | 22 | 14 | $6.28$ | $61.6$ | $61.2\pm 0.3$
full | 38 | 17 | $13.06$ | $90.6$ | $90.9\pm 0.1$
Table 9: Coverage fraction $f_{\rm in}=N_{\rm in}/N$ for $\gamma$ at its best measured value for 1-, 2-, and $3\sigma$ intervals, for the plug-in method and the simpler approach based on the profile likelihood. The quoted uncertainties are due to the limited number of pseudoexperiments. Combination | $\eta$ |  $\alpha$ (plug-in) |  $\alpha$ (profile likelihood)
---|---|---|---
$DK^{\pm}$ |  $0.6827$ $(1\sigma)$ | $0.6874\pm 0.0050$ | $0.6508\pm 0.0051$
|  $0.9545$ $(2\sigma)$ | $0.9543\pm 0.0023$ | $0.9414\pm 0.0025$
|  $0.9973$ $(3\sigma)$ | $0.9952\pm 0.0007$ | $0.9947\pm 0.0008$
$D\pi^{\pm}$ |  $0.6827$ $(1\sigma)$ | $0.5945\pm 0.0053$ | $0.5105\pm 0.0054$
|  $0.9545$ $(2\sigma)$ | $0.9391\pm 0.0026$ | $0.9238\pm 0.0029$
|  $0.9973$ $(3\sigma)$ | $0.9960\pm 0.0007$ | $0.9919\pm 0.0010$
$DK^{\pm}$ and $D\pi^{\pm}$ |  $0.6827$ $(1\sigma)$ | $0.6394\pm 0.0050$ | $0.5839\pm 0.0051$
|  $0.9545$ $(2\sigma)$ | $0.9374\pm 0.0025$ | $0.9112\pm 0.0030$
|  $0.9973$ $(3\sigma)$ | $0.9929\pm 0.0009$ | $0.9912\pm 0.0010$

For the two-body and four-body GLW/ADS analyses no information on systematic
correlations is available. Consequently, they are assumed to be zero in the
nominal combinations. Their possible influence is assessed by computing the
effect of a large number of random correlation matrices on the expected
confidence intervals. A maximum correlation of $75\%$ is considered in the
random matrices. The expected intervals are computed by generating
pseudodatasets at the best-fit points of the three combinations, and then, for
each pseudodataset, by computing its profile $\Delta\chi^{2}$ curve, and
taking the average of these curves. The $DK^{\pm}$ combination is unaffected.
The $D\pi^{\pm}$ combination, however, is affected to a large extent, as the
values of several observables are limited by systematic uncertainties.
Conservatively, the maximum of the $p$-values observed for all random
correlation matrices is considered. The nominal $1\sigma$ intervals are
asymmetrically enlarged by $12\%$ to match the maximum. The full combination
is only slightly affected. The systematic uncertainty is fully concentrated in
the lower side of the interval. Therefore, a systematic uncertainty of
$2.5^{\circ}$ $(5.0^{\circ})$ is added in quadrature to the lower $1\sigma$
$(2\sigma)$ errors.

The linearity of the combination procedure was checked by computing values for
all observables using the best-fit point of the full combination and the
relations from Section 2. Assuming the experimental covariances, the best-fit
point was perfectly reproduced, and the procedure was found to be unbiased.

In summary, the $DK^{\pm}$ combination does not require corrections. In case
of the $D\pi^{\pm}$ and full combinations, the intervals are enlarged to
account for both neglected systematic correlations and undercoverage.

##  6 Conclusion

A combination of recent LHCb results [25, 24, 26] is used to measure the CKM
angle $\gamma$. The decays $B^{\pm}\\!\rightarrow DK^{\pm}$ and
$B^{\pm}\\!\rightarrow D\pi^{\pm}$ are used, where the $D$ meson decays into
$K^{+}K^{-}$, $\pi^{+}\pi^{-}$, $K^{\pm}\pi^{\mp}$,
$K^{0}_{\rm\scriptscriptstyle S}\pi^{+}\pi^{-}$, $K^{0}_{\rm\scriptscriptstyle
S}K^{+}K^{-}$, or $K^{\pm}\pi^{\mp}\pi^{+}\pi^{\mp}$ final states. The effect
of $D^{0}$–$\kern 1.99997pt\overline{\kern-1.99997ptD}{}^{0}$ mixing is taken
into account in the ADS analysis of both $B^{\pm}\\!\rightarrow DK^{\pm}$ and
$B^{\pm}\\!\rightarrow D\pi^{\pm}$ decays. Using only $B^{\pm}\\!\rightarrow
DK^{\pm}$ results, a best-fit value in $[0,180]^{\circ}$ of
$\gamma=72.0^{\circ}$ is found and confidence intervals are set using a
frequentist procedure

$\displaystyle\gamma\in[56.4,86.7]^{\circ}\quad$ $\displaystyle{\rm
at~{}68\%\,CL}\,,$ $\displaystyle\gamma\in[42.6,99.6]^{\circ}\quad$
$\displaystyle{\rm at~{}95\%\,CL}\,.$

Taking the best-fit value as central value, the first interval is translated
to

$\gamma=(72.0^{+14.7}_{-15.6})^{\circ}\,{\rm at~{}68\%\,CL}\,.$

At $99\%$ CL a second (local) minimum contributes to the interval. When
combining results from $B^{\pm}\\!\rightarrow D\pi^{\pm}$ decays alone, a
best-fit value of $\gamma=18.9^{\circ}$ is found and the following confidence
intervals are set

$\displaystyle\gamma\in[7.4,99.2]^{\circ}\quad\cup\quad[167.9,176.4]^{\circ}\quad$
$\displaystyle{\rm at~{}68\%\,CL}\,,$

and no constraint is set at $95\%$ CL. For the first time, information from
$B^{\pm}\\!\rightarrow D\pi^{\pm}$ decays is included in a combination. When
these results are included, the best-fit value becomes $\gamma=72.6^{\circ}$
and the following confidence intervals are set

$\displaystyle\gamma\in[55.4,82.3]^{\circ}\quad$ $\displaystyle{\rm
at~{}68\%\,CL}\,,$ $\displaystyle\gamma\in[40.2,92.7]^{\circ}\quad$
$\displaystyle{\rm at~{}95\%\,CL}\,.$

All quoted values are modulo $180^{\circ}$. The coverage of our frequentist
method was evaluated and found to be exact when combining
$B^{\pm}\\!\rightarrow DK^{\pm}$ results alone, and accurate within 4% (2%) at
$1\sigma$ ($2\sigma$) when combining $B^{\pm}\\!\rightarrow DK^{\pm}$ and
$B^{\pm}\\!\rightarrow D\pi^{\pm}$ results. The final intervals have been
scaled up to account for this undercoverage, and to account for neglected
systematic correlations.

## 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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##  7 Supplemental material

The following material is to be made publicly available, but is not to be
included in the paper.

Figure 5: Graphs showing $1-{\rm CL}$ for $\delta_{B}^{K}$, $r_{B}^{K}$, and
$\gamma$, separately for the GLW/ADS (light green) and GGSZ (dark purple)
parts of the $DK^{\pm}$-only combination.

Figure 6: Profile likelihood contours, separately for the GGSZ (blue) and two-
body and four-body GLW/ADS (orange) parts of the $DK^{\pm}$ only combination.
The contours are the usual $n\sigma$ profile likelihood contours, where
$\Delta\chi^{2}=n^{2}$ with $n=1,2$. The markers correspond to the best-fit
points. Figure 7: Graphs showing $1-{\rm CL}$ for $\gamma$, separately for the
$DK^{\pm}$-only combination (dark purple) and $D\pi^{\pm}$-only combination
(light green).