EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)

 CERN-PH-EP-2014-082 LHCb-PAPER-2014-013 June 11, 2014

Measurement of $C\\!P$ asymmetry in $D^{0}\rightarrow K^{-}K^{+}$ and
$D^{0}\rightarrow\pi^{-}\pi^{+}$ decays

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

Time-integrated $C\\!P$ asymmetries in $D^{0}$ decays to the final states
$K^{-}K^{+}$ and $\pi^{-}\pi^{+}$ are measured using proton–proton collisions
corresponding to $3\mathrm{\,fb}^{-1}$ of integrated luminosity collected at
centre-of-mass energies of $7\mathrm{\,Te\kern-1.00006ptV}$ and
$8\mathrm{\,Te\kern-1.00006ptV}$. The $D^{0}$ mesons are produced in
semileptonic $b$-hadron decays, where the charge of the accompanying muon is
used to determine the initial flavour of the charm meson. The difference in
$C\\!P$ asymmetries between the two final states is measured to be

$\displaystyle\Delta
A_{C\\!P}=A_{C\\!P}(K^{-}K^{+})-A_{C\\!P}(\pi^{-}\pi^{+})=(+0.14\pm
0.16\mathrm{\,(stat)}\pm 0.08\mathrm{\,(syst)})\%\ .$

A measurement of $A_{C\\!P}(K^{-}K^{+})$ is obtained assuming negligible
$C\\!P$ violation in charm mixing and in Cabibbo-favoured $D$ decays. It is
found to be

$\displaystyle A_{C\\!P}(K^{-}K^{+})=(-0.06\pm 0.15\mathrm{\,(stat)}\pm
0.10\mathrm{\,(syst)})\%\ ,$

where the correlation coefficient between $\Delta A_{C\\!P}$ and
$A_{C\\!P}(K^{-}K^{+})$ is $\rho=0.28$. By combining these results, the
$C\\!P$ asymmetry in the $D^{0}\rightarrow\pi^{-}\pi^{+}$ channel is
$A_{C\\!P}(\pi^{-}\pi^{+})=(-0.20\pm 0.19\mathrm{\,(stat)}\pm
0.10\mathrm{\,(syst)})\%$.

Submitted to JHEP

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

LHCb collaboration

R. Aaij41, B. Adeva37, M. Adinolfi46, A. Affolder52, Z. Ajaltouni5, J.
Albrecht9, F. Alessio38, M. Alexander51, S. Ali41, G. Alkhazov30, P. Alvarez
Cartelle37, A.A. Alves Jr25,38, S. Amato2, S. Amerio22, Y. Amhis7, L. An3, L.
Anderlini17,g, J. Anderson40, R. Andreassen57, M. Andreotti16,f, J.E.
Andrews58, R.B. Appleby54, O. Aquines Gutierrez10, F. Archilli38, A.
Artamonov35, M. Artuso59, E. Aslanides6, G. Auriemma25,n, M. Baalouch5, S.
Bachmann11, J.J. Back48, A. Badalov36, V. Balagura31, W. Baldini16, R.J.
Barlow54, C. Barschel38, S. Barsuk7, W. Barter47, V. Batozskaya28, Th.
Bauer41, A. Bay39, L. Beaucourt4, J. Beddow51, F. Bedeschi23, I. Bediaga1, S.
Belogurov31, K. Belous35, I. Belyaev31, E. Ben-Haim8, G. Bencivenni18, S.
Benson38, J. Benton46, A. Berezhnoy32, R. Bernet40, M.-O. Bettler47, M. van
Beuzekom41, A. Bien11, S. Bifani45, T. Bird54, A. Bizzeti17,i, P.M.
Bjørnstad54, T. Blake48, F. Blanc39, J. Blouw10, S. Blusk59, V. Bocci25, A.
Bondar34, N. Bondar30,38, W. Bonivento15,38, S. Borghi54, A. Borgia59, M.
Borsato7, T.J.V. Bowcock52, E. Bowen40, C. Bozzi16, T. Brambach9, J. van den
Brand42, J. Bressieux39, D. Brett54, M. Britsch10, T. Britton59, J.
Brodzicka54, N.H. Brook46, H. Brown52, A. Bursche40, G. Busetto22,q, J.
Buytaert38, S. Cadeddu15, R. Calabrese16,f, M. Calvi20,k, M. Calvo Gomez36,o,
A. Camboni36, P. Campana18,38, D. Campora Perez38, A. Carbone14,d, G.
Carboni24,l, R. Cardinale19,38,j, A. Cardini15, H. Carranza-Mejia50, L.
Carson50, K. Carvalho Akiba2, G. Casse52, L. Cassina20, L. Castillo Garcia38,
M. Cattaneo38, Ch. Cauet9, R. Cenci58, M. Charles8, Ph. Charpentier38, S.
Chen54, S.-F. Cheung55, N. Chiapolini40, M. Chrzaszcz40,26, K. Ciba38, X. Cid
Vidal38, G. Ciezarek53, P.E.L. Clarke50, M. Clemencic38, H.V. Cliff47, J.
Closier38, V. Coco38, J. Cogan6, E. Cogneras5, P. Collins38, A. Comerma-
Montells11, A. Contu15,38, A. Cook46, M. Coombes46, S. Coquereau8, G. Corti38,
M. Corvo16,f, I. Counts56, B. Couturier38, G.A. Cowan50, D.C. Craik48, M. Cruz
Torres60, S. Cunliffe53, R. Currie50, C. D’Ambrosio38, J. Dalseno46, P.
David8, P.N.Y. David41, A. Davis57, K. De Bruyn41, S. De Capua54, M. De
Cian11, J.M. De Miranda1, L. De Paula2, W. De Silva57, P. De Simone18, D.
Decamp4, M. Deckenhoff9, L. Del Buono8, N. Déléage4, D. Derkach55, O.
Deschamps5, F. Dettori42, A. Di Canto38, H. Dijkstra38, S. Donleavy52, F.
Dordei11, M. Dorigo39, A. Dosil Suárez37, D. Dossett48, A. Dovbnya43, G.
Dujany54, F. Dupertuis39, P. Durante38, R. Dzhelyadin35, A. Dziurda26, A.
Dzyuba30, S. Easo49,38, U. Egede53, V. Egorychev31, S. Eidelman34, S.
Eisenhardt50, U. Eitschberger9, R. Ekelhof9, L. Eklund51,38, I. El Rifai5, Ch.
Elsasser40, S. Ely59, S. Esen11, T. Evans55, A. Falabella16,f, C. Färber11, C.
Farinelli41, N. Farley45, S. Farry52, D. Ferguson50, V. Fernandez Albor37, F.
Ferreira Rodrigues1, M. Ferro-Luzzi38, S. Filippov33, M. Fiore16,f, M.
Fiorini16,f, M. Firlej27, C. Fitzpatrick38, T. Fiutowski27, M. Fontana10, F.
Fontanelli19,j, R. Forty38, O. Francisco2, M. Frank38, C. Frei38, M.
Frosini17,38,g, J. Fu21,38, E. Furfaro24,l, A. Gallas Torreira37, D.
Galli14,d, S. Gallorini22, S. Gambetta19,j, M. Gandelman2, P. Gandini59, Y.
Gao3, J. Garofoli59, J. Garra Tico47, L. Garrido36, C. Gaspar38, R. Gauld55,
L. Gavardi9, E. Gersabeck11, M. Gersabeck54, T. Gershon48, Ph. Ghez4, A.
Gianelle22, S. Giani’39, V. Gibson47, L. Giubega29, V.V. Gligorov38, C.
Göbel60, D. Golubkov31, A. Golutvin53,31,38, A. Gomes1,a, H. Gordon38, C.
Gotti20, M. Grabalosa Gándara5, R. Graciani Diaz36, L.A. Granado Cardoso38, E.
Graugés36, G. Graziani17, A. Grecu29, E. Greening55, S. Gregson47, P.
Griffith45, L. Grillo11, O. Grünberg62, B. Gui59, E. Gushchin33, Yu. Guz35,38,
T. Gys38, C. Hadjivasiliou59, G. Haefeli39, C. Haen38, S.C. Haines47, S.
Hall53, B. Hamilton58, T. Hampson46, X. Han11, S. Hansmann-Menzemer11, N.
Harnew55, S.T. Harnew46, J. Harrison54, T. Hartmann62, J. He38, T. Head38, V.
Heijne41, K. Hennessy52, P. Henrard5, L. Henry8, J.A. Hernando Morata37, E.
van Herwijnen38, M. Heß62, A. Hicheur1, D. Hill55, M. Hoballah5, C. Hombach54,
W. Hulsbergen41, P. Hunt55, N. Hussain55, D. Hutchcroft52, D. Hynds51, M.
Idzik27, P. Ilten56, R. Jacobsson38, A. Jaeger11, J. Jalocha55, E. Jans41, P.
Jaton39, A. Jawahery58, M. Jezabek26, F. Jing3, M. John55, D. Johnson55, C.R.
Jones47, C. Joram38, B. Jost38, N. Jurik59, M. Kaballo9, S. Kandybei43, W.
Kanso6, M. Karacson38, T.M. Karbach38, M. Kelsey59, I.R. Kenyon45, T. Ketel42,
B. Khanji20, C. Khurewathanakul39, S. Klaver54, O. Kochebina7, M. Kolpin11, I.
Komarov39, R.F. Koopman42, P. Koppenburg41,38, M. Korolev32, A. Kozlinskiy41,
L. Kravchuk33, K. Kreplin11, M. Kreps48, G. Krocker11, P. Krokovny34, F.
Kruse9, M. Kucharczyk20,26,38,k, V. Kudryavtsev34, K. Kurek28, T.
Kvaratskheliya31, V.N. La Thi39, D. Lacarrere38, G. Lafferty54, A. Lai15, D.
Lambert50, R.W. Lambert42, E. Lanciotti38, G. Lanfranchi18, C. Langenbruch38,
B. Langhans38, T. Latham48, C. Lazzeroni45, R. Le Gac6, J. van Leerdam41,
J.-P. Lees4, R. Lefèvre5, A. Leflat32, J. Lefrançois7, S. Leo23, O. Leroy6, T.
Lesiak26, B. Leverington11, Y. Li3, M. Liles52, R. Lindner38, C. Linn38, F.
Lionetto40, B. Liu15, G. Liu38, S. Lohn38, I. Longstaff51, J.H. Lopes2, N.
Lopez-March39, P. Lowdon40, H. Lu3, D. Lucchesi22,q, H. Luo50, A. Lupato22, E.
Luppi16,f, O. Lupton55, F. Machefert7, I.V. Machikhiliyan31, F. Maciuc29, O.
Maev30, S. Malde55, G. Manca15,e, G. Mancinelli6, M. Manzali16,f, J. Maratas5,
J.F. Marchand4, U. Marconi14, C. Marin Benito36, P. Marino23,s, R. Märki39, J.
Marks11, G. Martellotti25, A. Martens8, A. Martín Sánchez7, M. Martinelli41,
D. Martinez Santos42, F. Martinez Vidal64, D. Martins Tostes2, A.
Massafferri1, R. Matev38, Z. Mathe38, C. Matteuzzi20, A. Mazurov16,f, M.
McCann53, J. McCarthy45, A. McNab54, R. McNulty12, B. McSkelly52, B.
Meadows57,55, F. Meier9, M. Meissner11, M. Merk41, D.A. Milanes8, M.-N.
Minard4, N. Moggi14, J. Molina Rodriguez60, S. Monteil5, D. Moran54, M.
Morandin22, P. Morawski26, A. Mordà6, M.J. Morello23,s, J. Moron27, A.-B.
Morris50, R. Mountain59, F. Muheim50, K. Müller40, R. Muresan29, M. Mussini14,
B. Muster39, P. Naik46, T. Nakada39, R. Nandakumar49, I. Nasteva2, M.
Needham50, N. Neri21, S. Neubert38, N. Neufeld38, M. Neuner11, A.D. Nguyen39,
T.D. Nguyen39, C. Nguyen-Mau39,p, M. Nicol7, V. Niess5, R. Niet9, N.
Nikitin32, T. Nikodem11, A. Novoselov35, A. Oblakowska-Mucha27, V.
Obraztsov35, S. Oggero41, S. Ogilvy51, O. Okhrimenko44, R. Oldeman15,e, G.
Onderwater65, M. Orlandea29, J.M. Otalora Goicochea2, P. Owen53, A.
Oyanguren64, B.K. Pal59, A. Palano13,c, F. Palombo21,t, M. Palutan18, J.
Panman38, A. Papanestis49,38, M. Pappagallo51, C. Parkes54, C.J. Parkinson9,
G. Passaleva17, G.D. Patel52, M. Patel53, C. Patrignani19,j, A. Pazos
Alvarez37, A. Pearce54, A. Pellegrino41, M. Pepe Altarelli38, S.
Perazzini14,d, E. Perez Trigo37, P. Perret5, M. Perrin-Terrin6, L.
Pescatore45, E. Pesen66, K. Petridis53, A. Petrolini19,j, E. Picatoste
Olloqui36, B. Pietrzyk4, T. Pilař48, D. Pinci25, A. Pistone19, S. Playfer50,
M. Plo Casasus37, F. Polci8, A. Poluektov48,34, E. Polycarpo2, A. Popov35, D.
Popov10, B. Popovici29, C. Potterat2, A. Powell55, J. Prisciandaro39, A.
Pritchard52, C. Prouve46, V. Pugatch44, A. Puig Navarro39, G. Punzi23,r, W.
Qian4, B. Rachwal26, J.H. Rademacker46, B. Rakotomiaramanana39, M. Rama18,
M.S. Rangel2, I. Raniuk43, N. Rauschmayr38, G. Raven42, S. Reichert54, M.M.
Reid48, A.C. dos Reis1, S. Ricciardi49, A. Richards53, M. Rihl38, K.
Rinnert52, V. Rives Molina36, D.A. Roa Romero5, P. Robbe7, A.B. Rodrigues1, E.
Rodrigues54, P. Rodriguez Perez54, S. Roiser38, V. Romanovsky35, A. Romero
Vidal37, M. Rotondo22, J. Rouvinet39, T. Ruf38, F. Ruffini23, H. Ruiz36, P.
Ruiz Valls64, G. Sabatino25,l, J.J. Saborido Silva37, N. Sagidova30, P.
Sail51, B. Saitta15,e, V. Salustino Guimaraes2, C. Sanchez Mayordomo64, B.
Sanmartin Sedes37, R. Santacesaria25, C. Santamarina Rios37, E.
Santovetti24,l, M. Sapunov6, A. Sarti18,m, C. Satriano25,n, A. Satta24, M.
Savrie16,f, D. Savrina31,32, M. Schiller42, H. Schindler38, M. Schlupp9, M.
Schmelling10, B. Schmidt38, O. Schneider39, A. Schopper38, M.-H. Schune7, R.
Schwemmer38, B. Sciascia18, A. Sciubba25, M. Seco37, A. Semennikov31, K.
Senderowska27, I. Sepp53, N. Serra40, J. Serrano6, L. Sestini22, P. Seyfert11,
M. Shapkin35, I. Shapoval16,43,f, Y. Shcheglov30, T. Shears52, L. Shekhtman34,
V. Shevchenko63, A. Shires9, R. Silva Coutinho48, G. Simi22, M. Sirendi47, N.
Skidmore46, T. Skwarnicki59, N.A. Smith52, E. Smith55,49, E. Smith53, J.
Smith47, M. Smith54, H. Snoek41, M.D. Sokoloff57, F.J.P. Soler51, F. Soomro39,
D. Souza46, B. Souza De Paula2, B. Spaan9, A. Sparkes50, F. Spinella23, P.
Spradlin51, F. Stagni38, S. Stahl11, O. Steinkamp40, O. Stenyakin35, S.
Stevenson55, S. Stoica29, S. Stone59, B. Storaci40, S. Stracka23,38, M.
Straticiuc29, U. Straumann40, R. Stroili22, V.K. Subbiah38, L. Sun57, W.
Sutcliffe53, K. Swientek27, S. Swientek9, V. Syropoulos42, M. Szczekowski28,
P. Szczypka39,38, D. Szilard2, T. Szumlak27, S. T’Jampens4, M. Teklishyn7, G.
Tellarini16,f, F. Teubert38, C. Thomas55, E. Thomas38, J. van Tilburg41, V.
Tisserand4, M. Tobin39, S. Tolk42, L. Tomassetti16,f, D. Tonelli38, S. Topp-
Joergensen55, N. Torr55, E. Tournefier4, S. Tourneur39, M.T. Tran39, M.
Tresch40, A. Tsaregorodtsev6, P. Tsopelas41, N. Tuning41, M. Ubeda Garcia38,
A. Ukleja28, A. Ustyuzhanin63, U. Uwer11, V. Vagnoni14, G. Valenti14, A.
Vallier7, R. Vazquez Gomez18, P. Vazquez Regueiro37, C. Vázquez Sierra37, S.
Vecchi16, J.J. Velthuis46, M. Veltri17,h, G. Veneziano39, M. Vesterinen11, B.
Viaud7, D. Vieira2, M. Vieites Diaz37, X. Vilasis-Cardona36,o, A. Vollhardt40,
D. Volyanskyy10, D. Voong46, A. Vorobyev30, V. Vorobyev34, C. Voß62, H.
Voss10, J.A. de Vries41, R. Waldi62, C. Wallace48, R. Wallace12, J. Walsh23,
S. Wandernoth11, J. Wang59, D.R. Ward47, N.K. Watson45, D. Websdale53, M.
Whitehead48, J. Wicht38, D. Wiedner11, G. Wilkinson55, M.P. Williams45, M.
Williams56, F.F. Wilson49, J. Wimberley58, J. Wishahi9, W. Wislicki28, M.
Witek26, G. Wormser7, S.A. Wotton47, S. Wright47, S. Wu3, K. Wyllie38, Y.
Xie61, Z. Xing59, Z. Xu39, Z. Yang3, X. Yuan3, O. Yushchenko35, M. Zangoli14,
M. Zavertyaev10,b, F. Zhang3, L. Zhang59, W.C. Zhang12, Y. Zhang3, A.
Zhelezov11, A. Zhokhov31, L. Zhong3, A. Zvyagin38.

1Centro Brasileiro de Pesquisas Físicas (CBPF), Rio de Janeiro, Brazil
2Universidade Federal do Rio de Janeiro (UFRJ), Rio de Janeiro, Brazil
3Center for High Energy Physics, Tsinghua University, Beijing, China
4LAPP, Université de Savoie, CNRS/IN2P3, Annecy-Le-Vieux, France
5Clermont Université, Université Blaise Pascal, CNRS/IN2P3, LPC, Clermont-
Ferrand, France
6CPPM, Aix-Marseille Université, CNRS/IN2P3, Marseille, France
7LAL, Université Paris-Sud, CNRS/IN2P3, Orsay, France
8LPNHE, Université Pierre et Marie Curie, Université Paris Diderot,
CNRS/IN2P3, Paris, France
9Fakultät Physik, Technische Universität Dortmund, Dortmund, Germany
10Max-Planck-Institut für Kernphysik (MPIK), Heidelberg, Germany
11Physikalisches Institut, Ruprecht-Karls-Universität Heidelberg, Heidelberg,
Germany
12School of Physics, University College Dublin, Dublin, Ireland
13Sezione INFN di Bari, Bari, Italy
14Sezione INFN di Bologna, Bologna, Italy
15Sezione INFN di Cagliari, Cagliari, Italy
16Sezione INFN di Ferrara, Ferrara, Italy
17Sezione INFN di Firenze, Firenze, Italy
18Laboratori Nazionali dell’INFN di Frascati, Frascati, Italy
19Sezione INFN di Genova, Genova, Italy
20Sezione INFN di Milano Bicocca, Milano, Italy
21Sezione INFN di Milano, Milano, Italy
22Sezione INFN di Padova, Padova, Italy
23Sezione INFN di Pisa, Pisa, Italy
24Sezione INFN di Roma Tor Vergata, Roma, Italy
25Sezione INFN di Roma La Sapienza, Roma, Italy
26Henryk Niewodniczanski Institute of Nuclear Physics Polish Academy of
Sciences, Kraków, Poland
27AGH - University of Science and Technology, Faculty of Physics and Applied
Computer Science, Kraków, Poland
28National Center for Nuclear Research (NCBJ), Warsaw, Poland
29Horia Hulubei National Institute of Physics and Nuclear Engineering,
Bucharest-Magurele, Romania
30Petersburg Nuclear Physics Institute (PNPI), Gatchina, Russia
31Institute of Theoretical and Experimental Physics (ITEP), Moscow, Russia
32Institute of Nuclear Physics, Moscow State University (SINP MSU), Moscow,
Russia
33Institute for Nuclear Research of the Russian Academy of Sciences (INR RAN),
Moscow, Russia
34Budker Institute of Nuclear Physics (SB RAS) and Novosibirsk State
University, Novosibirsk, Russia
35Institute for High Energy Physics (IHEP), Protvino, Russia
36Universitat de Barcelona, Barcelona, Spain
37Universidad de Santiago de Compostela, Santiago de Compostela, Spain
38European Organization for Nuclear Research (CERN), Geneva, Switzerland
39Ecole Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland
40Physik-Institut, Universität Zürich, Zürich, Switzerland
41Nikhef National Institute for Subatomic Physics, Amsterdam, The Netherlands
42Nikhef National Institute for Subatomic Physics and VU University Amsterdam,
Amsterdam, The Netherlands
43NSC Kharkiv Institute of Physics and Technology (NSC KIPT), Kharkiv, Ukraine
44Institute for Nuclear Research of the National Academy of Sciences (KINR),
Kyiv, Ukraine
45University of Birmingham, Birmingham, United Kingdom
46H.H. Wills Physics Laboratory, University of Bristol, Bristol, United
Kingdom
47Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom
48Department of Physics, University of Warwick, Coventry, United Kingdom
49STFC Rutherford Appleton Laboratory, Didcot, United Kingdom
50School of Physics and Astronomy, University of Edinburgh, Edinburgh, United
Kingdom
51School of Physics and Astronomy, University of Glasgow, Glasgow, United
Kingdom
52Oliver Lodge Laboratory, University of Liverpool, Liverpool, United Kingdom
53Imperial College London, London, United Kingdom
54School of Physics and Astronomy, University of Manchester, Manchester,
United Kingdom
55Department of Physics, University of Oxford, Oxford, United Kingdom
56Massachusetts Institute of Technology, Cambridge, MA, United States
57University of Cincinnati, Cincinnati, OH, United States
58University of Maryland, College Park, MD, United States
59Syracuse University, Syracuse, NY, United States
60Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), Rio de
Janeiro, Brazil, associated to 2
61Institute of Particle Physics, Central China Normal University, Wuhan,
Hubei, China, associated to 3
62Institut für Physik, Universität Rostock, Rostock, Germany, associated to 11
63National Research Centre Kurchatov Institute, Moscow, Russia, associated to
31
64Instituto de Fisica Corpuscular (IFIC), Universitat de Valencia-CSIC,
Valencia, Spain, associated to 36
65KVI - University of Groningen, Groningen, The Netherlands, associated to 41
66Celal Bayar University, Manisa, Turkey, associated to 38

aUniversidade Federal do Triângulo Mineiro (UFTM), Uberaba-MG, Brazil
bP.N. Lebedev Physical Institute, Russian Academy of Science (LPI RAS),
Moscow, Russia
cUniversità di Bari, Bari, Italy
dUniversità di Bologna, Bologna, Italy
eUniversità di Cagliari, Cagliari, Italy
fUniversità di Ferrara, Ferrara, Italy
gUniversità di Firenze, Firenze, Italy
hUniversità di Urbino, Urbino, Italy
iUniversità di Modena e Reggio Emilia, Modena, Italy
jUniversità di Genova, Genova, Italy
kUniversità di Milano Bicocca, Milano, Italy
lUniversità di Roma Tor Vergata, Roma, Italy
mUniversità di Roma La Sapienza, Roma, Italy
nUniversità della Basilicata, Potenza, Italy
oLIFAELS, La Salle, Universitat Ramon Llull, Barcelona, Spain
pHanoi University of Science, Hanoi, Viet Nam
qUniversità di Padova, Padova, Italy
rUniversità di Pisa, Pisa, Italy
sScuola Normale Superiore, Pisa, Italy
tUniversità degli Studi di Milano, Milano, Italy

##  1 Introduction

Decays of charm mesons mediated by the weak interaction provide an attractive
testing ground for physics beyond the Standard Model (SM). Violations of
charge-parity ($C\\!P$) symmetry are predicted to be small in charm decays,
but could be enhanced in the presence of non-SM physics. Direct $C\\!P$
violation arises when two or more amplitudes with different weak and strong
phases contribute to the same final state. This is possible in singly Cabibbo-
suppressed ${{D}^{0}}\\!\rightarrow{{K}^{-}}{{K}^{+}}$ and
${{D}^{0}}\\!\rightarrow{{\pi}^{-}}{{\pi}^{+}}$ decays,111The inclusion of
charge-conjugate processes is implied throughout this paper, unless explicitly
stated otherwise. where significant penguin contributions can be expected [1].
Under SU(3) flavour symmetry, which is approximately valid in heavy quark
transitions, the direct $C\\!P$ asymmetries in these decays are expected to
have equal magnitudes and opposite sign. For a long time, direct $C\\!P$
violation in these decays was expected to be below the $10^{-3}$ level [2];
however, this prediction has been revisited recently and asymmetries at a few
times $10^{-3}$ cannot be excluded within the SM [3, 4, 5, 6]. Indirect
$C\\!P$ violation, occurring through ${D}^{0}$ mixing, is expected to be
negligible at the current experimental precision [7, 2] and measured to be
consistent with zero [8]. To date, $C\\!P$ violation in charm decays has not
been established experimentally.

In this paper the $C\\!P$ asymmetries in
${{D}^{0}}\\!\rightarrow{{K}^{-}}{{K}^{+}}$ and
${{D}^{0}}\\!\rightarrow{{\pi}^{-}}{{\pi}^{+}}$ decays are measured in
semileptonic $b$-hadron decays using the muon charge to identify (tag) the
flavour of the ${D}^{0}$ meson at production. This time-integrated $C\\!P$
asymmetry receives contributions from both direct and indirect $C\\!P$
violation. The difference of these asymmetries ($\Delta A_{{C\\!P}}$) was
measured at LHCb [9, *LHCb-CONF-2013-003, 11]. Assuming indirect $C\\!P$
violation to be independent of the decay mode [7, 2], only the effect of
direct $C\\!P$ violation remains in $\Delta A_{{C\\!P}}$. This paper
supersedes the previous $\Delta A_{{C\\!P}}$ result from Ref. [11] that was
based on one third of the data. In this paper the individual $C\\!P$
asymmetries in ${{D}^{0}}\\!\rightarrow{{\pi}^{-}}{{\pi}^{+}}$ and
${{D}^{0}}\\!\rightarrow{{K}^{-}}{{K}^{+}}$ decays are also measured, using
samples of Cabibbo-favoured ${D}^{0}$ and ${D}^{+}$ decays to correct for
spurious asymmetries due to detection and production effects. Individual
$C\\!P$ asymmetries and their difference have been measured by several other
experiments [12, 13, *Collaboration:2012qw, 15, *Ko:2012px], all using
${D}^{0}$ mesons tagged by the charge of the pion from
${{D}^{*+}}\rightarrow{{D}^{0}}{{\pi}^{+}}$ decays. The world average values
[8] are $A_{{C\\!P}}({{K}^{-}}{{K}^{+}})=(-0.15\pm 0.14)\%$ and
$A_{{C\\!P}}({{\pi}^{-}}{{\pi}^{+}})=(0.18\pm 0.15)\%$ for the individual
asymmetries and $\Delta A_{{C\\!P}}=(-0.33\pm 0.12)\%$ for the difference.

##  2 Method and formalism

Our procedure to measure the difference in $C\\!P$ asymmetries, $\Delta
A_{{C\\!P}}$, follows Ref. [11]. The observed (raw) asymmetry for a $D$ meson
decay rate to a final state $f$ is defined as

$A_{\rm raw}\equiv\frac{N({D}\rightarrow f)-N({{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}}\rightarrow\overline{f})}{N({D}\rightarrow
f)+N({{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}}\rightarrow\overline{f})}\ ,$ (1)

where $N$ is the number of observed decays, $D$ is either a ${D}^{+}$ or
${D}^{0}$ meson, and ${\kern 1.99997pt\overline{\kern-1.99997ptD}{}}$ is
either a ${D}^{-}$ or ${\kern 1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}$
meson. For decays to a $C\\!P$ eigenstate, where $f=\overline{f}$, the initial
flavour of the ${D}^{0}$ meson is tagged by the charge of the accompanying
muon in the semileptonic decay ${\kern
1.79993pt\overline{\kern-1.79993ptB}{}}\rightarrow{{D}^{0}}{\mu^{-}}{{\overline{\nu}}_{\mu}}X$,
where $X$ denotes possible other particles produced in the decay. Neglecting
third-order terms in the asymmetries, the raw asymmetry in the decays
${{D}^{0}}\\!\rightarrow{{K}^{-}}{{K}^{+}}$ and
${{D}^{0}}\\!\rightarrow{{\pi}^{-}}{{\pi}^{+}}$ is

$A_{\rm raw}=A_{{C\\!P}}+A_{D}({\mu^{-}})+A_{P}({\kern
1.79993pt\overline{\kern-1.79993ptB}{}})\ ,$ (2)

where $A_{D}({\mu^{-}})$ is any charge-dependent asymmetry in muon
reconstruction efficiency and $A_{P}({\kern
1.79993pt\overline{\kern-1.79993ptB}{}})$ is the asymmetry between the numbers
of $b$ and $\overline{b}$ hadrons (denoted as $\kern
1.79993pt\overline{\kern-1.79993ptB}{}$ and $B$, respectively) produced in the
LHCb acceptance, which includes possible $C\\!P$ violation in ${B}^{0}$
mixing. The production and detection asymmetries are common to the
${{D}^{0}}\\!\rightarrow{{K}^{-}}{{K}^{+}}$ and
${{D}^{0}}\\!\rightarrow{{\pi}^{-}}{{\pi}^{+}}$ decay modes, so they cancel in
the difference of the raw asymmetries, giving

$\Delta A_{{C\\!P}}=A_{\rm raw}({{K}^{-}}{{K}^{+}})-A_{\rm
raw}({{\pi}^{-}}{{\pi}^{+}})=A_{{C\\!P}}({{K}^{-}}{{K}^{+}})-A_{{C\\!P}}({{\pi}^{-}}{{\pi}^{+}})\
.$ (3)

The production and muon detection asymmetry in Eq. (2) can also be removed
using the Cabibbo-favoured ${{D}^{0}}\\!\rightarrow{{K}^{-}}{{\pi}^{+}}$ decay
mode in ${\kern
1.79993pt\overline{\kern-1.79993ptB}{}}\rightarrow{{D}^{0}}{\mu^{-}}{{\overline{\nu}}_{\mu}}X$
decays. In this decay, $C\\!P$ violation can be neglected as it is expected to
be significantly suppressed compared to our sensitivity for measuring $C\\!P$
violation in ${{D}^{0}}\\!\rightarrow{{K}^{-}}{{K}^{+}}$ and
${{D}^{0}}\\!\rightarrow{{\pi}^{-}}{{\pi}^{+}}$ decays. In the reconstruction
of the ${{K}^{-}}{{\pi}^{+}}$ final state there is an instrumental asymmetry,
$A_{D}({{K}^{-}}{{\pi}^{+}})$, due to the different interaction cross section
of positively and negatively charged kaons in the detector material. Also,
other detector-related effects, for example due to the acceptance, selection
and detection inefficiencies, can contribute to this detection asymmetry. The
raw asymmetry in this decay mode is then

$A_{\rm raw}({{K}^{-}}{{\pi}^{+}})=A_{D}({\mu^{-}})+A_{P}({\kern
1.79993pt\overline{\kern-1.79993ptB}{}})+A_{D}({{K}^{-}}{{\pi}^{+}})\ .$ (4)

The detection asymmetry $A_{D}({{K}^{-}}{{\pi}^{+}})$ of the final state
${{K}^{-}}{{\pi}^{+}}$ is obtained from ${D}^{+}$ decays produced directly in
$pp$ collisions (so-called prompt ${D}^{+}$ decays). Two decay modes are used,
${{D}^{+}}\\!\rightarrow{{K}^{-}}{{\pi}^{+}}{{\pi}^{+}}$ and
${{D}^{+}}\\!\rightarrow{{\kern
1.99997pt\overline{\kern-1.99997ptK}{}}{}^{0}}{{\pi}^{+}}$ with ${{\kern
1.99997pt\overline{\kern-1.99997ptK}{}}{}^{0}}\rightarrow{{\pi}^{+}}{{\pi}^{-}}$.
The raw asymmetry of ${{D}^{+}}\\!\rightarrow{{K}^{-}}{{\pi}^{+}}{{\pi}^{+}}$
decays is

$A_{\rm
raw}({{K}^{-}}{{\pi}^{+}}{{\pi}^{+}})=A_{P}({{D}^{+}})+A_{D}({{K}^{-}}{{\pi}^{+}})+A_{D}({{\pi}^{+}})\
,$ (5)

where $A_{P}({{D}^{+}})$ is the production asymmetry of prompt ${D}^{+}$
mesons and $A_{D}({{\pi}^{+}})$ is the detection asymmetry of the other
charged pion. The raw asymmetry of ${{D}^{+}}\\!\rightarrow{{\kern
1.99997pt\overline{\kern-1.99997ptK}{}}{}^{0}}{{\pi}^{+}}$ decays is

$A_{\rm raw}({{\kern
1.99997pt\overline{\kern-1.99997ptK}{}}{}^{0}}{{\pi}^{+}})=A_{P}({{D}^{+}})+A_{D}({{\pi}^{+}})-A_{D}({{K}^{0}})\
,$ (6)

where $A_{D}({{K}^{0}})$ is the detection asymmetry of the decay
${{K}^{0}}\\!\rightarrow{{\pi}^{+}}{{\pi}^{-}}$, which is discussed later.
Taking the difference between Eqs. (5) and (6), $A_{D}({{K}^{-}}{{\pi}^{+}})$
is obtained as

$A_{D}({{K}^{-}}{{\pi}^{+}})=A_{\rm
raw}({{K}^{-}}{{\pi}^{+}}{{\pi}^{+}})-A_{\rm raw}({{\kern
1.99997pt\overline{\kern-1.99997ptK}{}}{}^{0}}{{\pi}^{+}})-A_{D}({{K}^{0}})\
.$ (7)

This method assumes negligible $C\\!P$ violation in these Cabibbo-favoured
${D}^{+}$ decay modes. By combining Eqs. (2) and (4), the $C\\!P$ asymmetry in
the ${{D}^{0}}\\!\rightarrow{{K}^{-}}{{K}^{+}}$ decay becomes

$A_{{C\\!P}}({{K}^{-}}{{K}^{+}})=A_{\rm raw}({{K}^{-}}{{K}^{+}})-A_{\rm
raw}({{K}^{-}}{{\pi}^{+}})+A_{D}({{K}^{-}}{{\pi}^{+}})\ ,$ (8)

where $A_{D}({{K}^{-}}{{\pi}^{+}})$ is taken from Eq. (7). The $C\\!P$
asymmetry in the ${{D}^{0}}\\!\rightarrow{{\pi}^{-}}{{\pi}^{+}}$ decay is
determined from the difference between the measurements of
$A_{{C\\!P}}({{K}^{-}}{{K}^{+}})$ and $\Delta A_{{C\\!P}}$.

##  3 Detector

The LHCb detector [17] is a single-arm forward spectrometer covering the
pseudorapidity range $2<\eta<5$, designed for the study of particles
containing $b$ or $c$ quarks. The detector includes a high-precision tracking
system consisting of a silicon-strip vertex detector surrounding the $pp$
interaction region, a large-area silicon-strip detector located upstream of a
dipole magnet with a bending power of about $4{\rm\,Tm}$, and three stations
of silicon-strip detectors and straw drift tubes placed downstream of the
magnet. The polarity of the magnetic field is regularly reversed during data
taking. The combined tracking system provides a momentum measurement with
relative uncertainty that varies from 0.4% at low momentum, $p$, to 0.6% at
100${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$, and impact parameter resolution of
20${\,\upmu\rm m}$ for charged particles with large transverse momentum,
$p_{\rm T}$. Different types of charged hadrons are distinguished by
information from two ring-imaging Cherenkov detectors [18]. Photon, electron
and hadron candidates are identified by a calorimeter system consisting of
scintillating-pad and preshower detectors, an electromagnetic calorimeter and
a hadronic calorimeter. Muons are identified by a system composed of
alternating layers of iron and multiwire proportional chambers. The trigger
[19] consists of a hardware stage, based on information from the calorimeter
and muon systems, followed by a two-stage software stage, that applies a full
event reconstruction.

##  4 Data set and selection

This analysis uses the data set collected by LHCb corresponding to an
integrated luminosity of $3\mbox{\,fb}^{-1}$. The data in 2011
($1\mbox{\,fb}^{-1}$) were taken at a centre-of-mass energy of
7$\mathrm{\,Te\kern-1.00006ptV}$ and the data in 2012 ($2\mbox{\,fb}^{-1}$)
were taken at a centre-of-mass energy of 8$\mathrm{\,Te\kern-1.00006ptV}$. The
fraction of data collected with up (down) polarity of the magnetic field is
$40\%$ ($60\%$) in 2011 and $52\%$ ($48\%$) in 2012\. Charge-dependent
detection asymmetries originating from any left-right asymmetry in the
detector change sign when the field polarity is reversed. By design, the
analysis method does not rely on any cancellation due to the regular field
reversals, since all detection asymmetries are already removed in the
determination of the $C\\!P$ asymmetries. This assumption is tested by
performing the analysis separately for the two polarities. To ensure that any
residual detection asymmetries cancel, the raw asymmetries are determined from
the arithmetic mean of the results obtained for the two magnet polarities.
Similarly, the analysis is performed separately for the 2011 and 2012 data as
detection asymmetries and production asymmetries change due to different
operational conditions.

At the hardware trigger stage, the events in the semileptonic $B$ decay modes
are required to be triggered by the muon system. The muon transverse momentum
must be larger than $1.64{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ for the 2011
data and larger than $1.76{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ for the 2012
data. In the software trigger, the muon candidate is first required to have
$\mbox{$p_{\rm T}$}>1.0{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ and a large
impact parameter. Then, the muon and one or two of the ${D}^{0}$ decay
products are required to be consistent with the topological signature of
$b$-hadron decays [19]. Since the $\Delta A_{{C\\!P}}$ measurement has no
detection asymmetry coming from the ${D}^{0}$ decay products, $B$ candidates
triggered on the presence of a final-state particle with high $p_{\rm T}$ and
large impact parameter are also accepted in the corresponding selection.

The remaining selection of semileptonic $B$ decays reduces the background from
prompt ${D}^{0}$ decays to the per-cent level. The residual background
consists mainly of combinations from inclusive $b$-hadron decays with other
particles in the event. The selection is similar to that in the previous
publication [11], except for the looser particle identification requirements
of the kaon candidates. To reduce the large
${{D}^{0}}\\!\rightarrow{{K}^{-}}{{\pi}^{+}}$ sample size only half of the
candidates (randomly selected) is kept.

In order to reduce possible biases induced by trigger criteria, the events in
the prompt charm decay modes are selected by the hardware trigger,
independently of the presence of the ${D}^{+}$ candidate. In the software
trigger, one of the final-state pions is first required to have $\mbox{$p_{\rm
T}$}>1.6{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ and a large impact parameter to
any primary vertex. This ensures that the distributions of the kaon and the
other pion in the ${{D}^{+}}\\!\rightarrow{{K}^{-}}{{\pi}^{+}}{{\pi}^{+}}$
decay and of the ${\kern 1.99997pt\overline{\kern-1.99997ptK}{}}{}^{0}$ meson
in the ${{D}^{+}}\\!\rightarrow{{\kern
1.99997pt\overline{\kern-1.99997ptK}{}}{}^{0}}{{\pi}^{+}}$ decay are not
biased by these trigger requirements. Finally, an exclusive selection is
applied for each ${D}^{+}$ decay mode in the last stage of the software
trigger. This is similar to the offline selection, where a secondary vertex is
reconstructed and required to be significantly displaced from any primary
vertex.

All particles are required to have $p>2{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$
and $\mbox{$p_{\rm T}$}>250{\mathrm{\,Me\kern-1.00006ptV\\!/}c}$.
Additionally, all tracks in the
${{D}^{+}}\\!\rightarrow{{K}^{-}}{{\pi}^{+}}{{\pi}^{+}}$ and
${{D}^{+}}\\!\rightarrow{{\kern
1.99997pt\overline{\kern-1.99997ptK}{}}{}^{0}}{{\pi}^{+}}$ decays are required
to have a large impact parameter with respect to any primary vertex. The
particle identification requirements are the same as those in the
${{D}^{0}}\\!\rightarrow{{K}^{-}}{{\pi}^{+}}$ decay mode from semileptonic $B$
decays. The neutral kaon in the ${{D}^{+}}\\!\rightarrow{{\kern
1.99997pt\overline{\kern-1.99997ptK}{}}{}^{0}}{{\pi}^{+}}$ decay is detected
in the ${{\pi}^{+}}{{\pi}^{-}}$ final state, which is dominated by the decay
of the ${K}^{0}_{\rm\scriptscriptstyle S}$ state. When
${K}^{0}_{\rm\scriptscriptstyle S}$ mesons decay early such that both pions
leave sufficiently many hits in the vertex detector and in the three
downstream tracking stations, the pions can be reconstructed as so-called long
tracks. When ${K}^{0}_{\rm\scriptscriptstyle S}$ mesons decay later such that
both pions do not leave enough hits in the vertex detector, but enough hits in
the rest of the tracking system, the pions can be reconstructed as so-called
downstream tracks. Downstream ${K}^{0}_{\rm\scriptscriptstyle S}$ candidates
are available only in 2012 data, since no dedicated trigger was available to
select these decays in 2011. For this reason, only
${{D}^{+}}\\!\rightarrow{{\kern
1.99997pt\overline{\kern-1.99997ptK}{}}{}^{0}}{{\pi}^{+}}$ decays formed with
long ${K}^{0}_{\rm\scriptscriptstyle S}$ candidates are used for the asymmetry
measurement. The downstream ${K}^{0}_{\rm\scriptscriptstyle S}$ candidates are
used to check the effect of the ${K}^{0}$ detection asymmetry. There is no
$p_{\rm T}$ requirement for the pions from downstream
${K}^{0}_{\rm\scriptscriptstyle S}$ candidates. All
${K}^{0}_{\rm\scriptscriptstyle S}$ candidates are required to have a large
impact parameter. Both ${D}^{+}$ and ${K}^{0}_{\rm\scriptscriptstyle S}$
candidates are required to have $\mbox{$p_{\rm
T}$}>1{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ and an accurately reconstructed
decay vertex. For the ${{D}^{+}}\\!\rightarrow{{K}^{-}}{{\pi}^{+}}{{\pi}^{+}}$
decay the scalar $p_{\rm T}$ sum of the ${D}^{+}$ daughters is required to be
larger than $2.8{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$. The ${D}^{+}$
candidates should have a large impact parameter and significant flight
distance from the primary vertex. Given the large branching fraction of the
${{D}^{+}}\\!\rightarrow{{K}^{-}}{{\pi}^{+}}{{\pi}^{+}}$ decay, only one fifth
of the available data set (randomly selected) is considered in the following.

To improve the mass resolution, a vertex fit [20] of the ${D}^{+}$ decay
products is made, where the ${D}^{+}$ candidate is constrained to originate
from the corresponding primary vertex. Additionally, in the decay
${{D}^{+}}\\!\rightarrow{{\kern
1.99997pt\overline{\kern-1.99997ptK}{}}{}^{0}}{{\pi}^{+}}$ the mass of the
${\kern 1.99997pt\overline{\kern-1.99997ptK}{}}{}^{0}$ meson is constrained to
the nominal value [21]. The momentum of all particles is corrected [22] to
improve the stability of the mass scale versus data taking period and to
reduce the width of the mass distribution.

##  5 Determination of the asymmetries

In this section the raw asymmetries are obtained from fits to the invariant
mass distributions. These numbers are corrected for effects coming from the
${K}^{0}$ detection asymmetry and wrong flavour tags. The contributions from
direct and indirect $C\\!P$ violation are determined and finally the $C\\!P$
asymmetries are calculated.

###  5.1 Invariant mass distributions

Invariant mass distributions for the ${D}^{0}$ and ${D}^{+}$ candidates are
shown in Fig. 1 with the fit results overlaid. For all decay modes the signal
is modelled by the sum of two Gaussian functions with common mean and a power-
law tail. For decays to non-$C\\!P$ eigenstates (i.e.,
${{D}^{0}}\\!\rightarrow{{K}^{-}}{{\pi}^{+}}$,
${{D}^{+}}\\!\rightarrow{{K}^{-}}{{\pi}^{+}}{{\pi}^{+}}$,
${{D}^{+}}\\!\rightarrow{{\kern
1.99997pt\overline{\kern-1.99997ptK}{}}{}^{0}}{{\pi}^{+}}$) different means
and average widths are allowed between $D$ and ${\kern
1.99997pt\overline{\kern-1.99997ptD}{}}$ states, due to a known charge-
dependent bias in the measurement of the momentum. The background is described
by an exponential function, with different slopes for $D$ and ${\kern
1.99997pt\overline{\kern-1.99997ptD}{}}$ states. An overall asymmetry in the
number of background events is also included in the model. The background from
misidentified ${{D}^{0}}\\!\rightarrow{{K}^{-}}{{\pi}^{+}}$ decays in the
${{D}^{0}}\\!\rightarrow{{\pi}^{-}}{{\pi}^{+}}$ invariant mass distribution is
modelled with a single Gaussian function with the same shape for both muon
tags and an additional asymmetry parameter. The numbers of signal decays
determined from fits to the invariant mass distributions are given in Table 1.

Table 1: Number of signal decays determined from fits to the invariant mass distributions. Decay sample | Signal decays
---|---
${{D}^{0}}\\!\rightarrow{{\pi}^{-}}{{\pi}^{+}}$ from $B$ | 773 541
${{D}^{0}}\\!\rightarrow{{K}^{-}}{{K}^{+}}$ from $B$, $\Delta A_{{C\\!P}}$ selection | 2 166 045
${{D}^{0}}\\!\rightarrow{{K}^{-}}{{K}^{+}}$ from $B$, $A_{{C\\!P}}({{K}^{-}}{{K}^{+}})$ selection | 1 821 462
${{D}^{0}}\\!\rightarrow{{K}^{-}}{{\pi}^{+}}$ from $B$ | 9 088 675
Prompt ${{D}^{+}}\\!\rightarrow{{K}^{-}}{{\pi}^{+}}{{\pi}^{+}}$ | 40 782 645
Prompt ${{D}^{+}}\\!\rightarrow{{\kern 1.99997pt\overline{\kern-1.99997ptK}{}}{}^{0}}{{\pi}^{+}}$, long ${K}^{0}_{\rm\scriptscriptstyle S}$ | 3 765 530
Prompt ${{D}^{+}}\\!\rightarrow{{\kern 1.99997pt\overline{\kern-1.99997ptK}{}}{}^{0}}{{\pi}^{+}}$, downstream ${K}^{0}_{\rm\scriptscriptstyle S}$ | 2 512 615

(a)(b)

(c)(d)

(e)

Figure 1: Invariant mass distributions for muon-tagged (a)
${{D}^{0}}\\!\rightarrow{{K}^{-}}{{K}^{+}}$, (b)
${{D}^{0}}\\!\rightarrow{{\pi}^{-}}{{\pi}^{+}}$ and (c)
${{D}^{0}}\\!\rightarrow{{K}^{-}}{{\pi}^{+}}$ candidates and for prompt (d)
${{D}^{+}}\\!\rightarrow{{K}^{-}}{{\pi}^{+}}{{\pi}^{+}}$ and (e)
${{D}^{+}}\\!\rightarrow{{\kern
1.99997pt\overline{\kern-1.99997ptK}{}}{}^{0}}{{\pi}^{+}}$ candidates. The
results of the fits are overlaid.

###  5.2 Differences in kinematic distributions

Production and detection asymmetries depend on the kinematic distributions of
the particles involved. Since the momentum distributions of the particles in
the signal and calibration decay modes are different, small residual
production and detection asymmetries can remain in the calculation of $\Delta
A_{{C\\!P}}$ and $A_{{C\\!P}}({{K}^{-}}{{K}^{+}})$. This effect is mitigated
by assigning weights to each candidate such that the kinematic distributions
are equalised. For the measurement of $\Delta A_{{C\\!P}}$, the
${{D}^{0}}\\!\rightarrow{{K}^{-}}{{K}^{+}}$ candidates are weighted according
to the $p_{\rm T}$ and $\eta$ values of the ${D}^{0}$, which are the kinematic
variables showing the most significant differences. The weights are chosen
such that the weighted and background-subtracted distributions of the
${D}^{0}$ and muon candidates agree with the corresponding (unweighted)
distributions in the ${{D}^{0}}\\!\rightarrow{{\pi}^{-}}{{\pi}^{+}}$ sample.
After weighting, the effective sample size is given by $N_{\rm
eff}=(\sum_{i=1}^{N}w_{i})^{2}/(\sum_{i=1}^{N}w_{i}^{2})$, where $w_{i}$ is
the weight of candidate $i$ and $N$ the total number of candidates. Due to the
good agreement in the kinematic distributions between the two decays, this
procedure reduces the statistical power of the weighted
${{D}^{0}}\\!\rightarrow{{K}^{-}}{{K}^{+}}$ event sample by only $8\%$.

For the measurement of $A_{{C\\!P}}({{K}^{-}}{{K}^{+}})$, additional weighting
steps for the ${D}^{+}$ calibration decay modes are needed. In the first step,
the ${{D}^{0}}\\!\rightarrow{{K}^{-}}{{\pi}^{+}}$ candidates are weighted
based on the $p_{\rm T}$ and $\eta$ values of the ${D}^{0}$ meson, such that
they agree with the corresponding (unweighted) distributions of the
${{D}^{0}}\\!\rightarrow{{K}^{-}}{{K}^{+}}$ candidates. This step, which
reduces the statistical power of the
${{D}^{0}}\\!\rightarrow{{K}^{-}}{{\pi}^{+}}$ sample by $3\%$, ensures the
cancellation of the $B$ production asymmetry and muon detection asymmetry. In
the second step, the ${{D}^{+}}\\!\rightarrow{{K}^{-}}{{\pi}^{+}}{{\pi}^{+}}$
candidates are weighted according to the $p_{\rm T}$ and $\eta$ values of both
the kaon and the pion that was not selected by the software trigger. This step
equalises the kinematic distributions of the ${K}^{-}$ and ${\pi}^{+}$ to
those of the (now weighted) ${{D}^{0}}\\!\rightarrow{{K}^{-}}{{\pi}^{+}}$
decay to ensure cancellation of the ${{K}^{-}}{{\pi}^{+}}$ detection
asymmetry. The resulting $50\%$ reduction in statistical power does not
contribute to the final uncertainty given the large number of
${{D}^{+}}\\!\rightarrow{{K}^{-}}{{\pi}^{+}}{{\pi}^{+}}$ candidates available.
In the last step, the ${{D}^{+}}\\!\rightarrow{{\kern
1.99997pt\overline{\kern-1.99997ptK}{}}{}^{0}}{{\pi}^{+}}$ candidates are
weighted according to the $p_{\rm T}$ and $\eta$ values of both the pion and
the ${D}^{+}$ candidate, such that they agree with the corresponding
(weighted) distributions of the
${{D}^{+}}\\!\rightarrow{{K}^{-}}{{\pi}^{+}}{{\pi}^{+}}$ candidates. The last
step ensures cancellation of the ${D}^{+}$ production asymmetry and detection
asymmetry from the pion that is used in the software trigger and reduces the
statistical power of the ${{D}^{+}}\\!\rightarrow{{\kern
1.99997pt\overline{\kern-1.99997ptK}{}}{}^{0}}{{\pi}^{+}}$ sample by $77\%$.

###  5.3 ${K}^{0}$ asymmetry

An asymmetry in the detection of a ${K}^{0}$ to the ${{\pi}^{+}}{{\pi}^{-}}$
final state arises from the combined effect of $C\\!P$ violation and mixing in
the neutral kaon system and the different interaction rates of ${K}^{0}$ and
${\kern 1.99997pt\overline{\kern-1.99997ptK}{}}{}^{0}$ in the detector
material. Due to material interactions, a pure ${K}^{0}_{\rm\scriptscriptstyle
L}$ state can change back into a superposition of
${K}^{0}_{\rm\scriptscriptstyle L}$ and ${K}^{0}_{\rm\scriptscriptstyle S}$
states [23]. These regeneration and $C\\!P$-violating effects are of the same
order and same sign in LHCb. To estimate the total ${K}^{0}$ detection
asymmetry, the mixing, $C\\!P$ violation and absorption in material need to be
described coherently. The amplitudes in the ${K}^{0}_{\rm\scriptscriptstyle
L}$ and ${K}^{0}_{\rm\scriptscriptstyle S}$ basis of an arbitrary neutral kaon
state in matter evolve as [24]

$\displaystyle{\alpha_{\rm\scriptscriptstyle L}}(t)$
$\displaystyle=\textrm{e}^{-i\,t\,\Sigma}\left[{\alpha_{\rm\scriptscriptstyle
L}}(0)\cos{(\Omega t)}-i\frac{{\alpha_{\rm\scriptscriptstyle
L}}(0)\Delta\lambda+{\alpha_{\rm\scriptscriptstyle
S}}(0)\Delta\chi}{2\Omega}\sin{(\Omega t)}\right]\ ,$ (9)
$\displaystyle{\alpha_{\rm\scriptscriptstyle S}}(t)$
$\displaystyle=\textrm{e}^{-i\,t\,\Sigma}\left[{\alpha_{\rm\scriptscriptstyle
S}}(0)\cos{(\Omega t)}+i\frac{{\alpha_{\rm\scriptscriptstyle
S}}(0)\Delta\lambda-{\alpha_{\rm\scriptscriptstyle
L}}(0)\Delta\chi}{2\Omega}\sin{(\Omega t)}\right]\ ,$ (10)

where the constants
$\Omega\equiv\frac{1}{2}\sqrt{\Delta\lambda^{2}+\Delta\chi^{2}}$ and
$\Sigma\equiv\frac{1}{2}({\lambda_{\rm\scriptscriptstyle
L}}+{\lambda_{\rm\scriptscriptstyle S}}+\chi+\bar{\chi})$ are given by the
masses ${m_{\rm\scriptscriptstyle L,S}}$ and decay widths
${\Gamma_{\rm\scriptscriptstyle L,S}}$ of the ${K}^{0}_{\rm\scriptscriptstyle
L}$ and ${K}^{0}_{\rm\scriptscriptstyle S}$ states and by the absorption
$\chi$ ($\bar{\chi}$) of ${K}^{0}$ (${\kern
1.99997pt\overline{\kern-1.99997ptK}{}}{}^{0}$) states through

$\displaystyle\Delta\lambda$ $\displaystyle={\lambda_{\rm\scriptscriptstyle
L}}-{\lambda_{\rm\scriptscriptstyle S}}=\Delta
m-\frac{i}{2}\Delta\Gamma=({m_{\rm\scriptscriptstyle
L}}-{m_{\rm\scriptscriptstyle S}})-\frac{i}{2}({\Gamma_{\rm\scriptscriptstyle
L}}-{\Gamma_{\rm\scriptscriptstyle S}})\ ,$ $\displaystyle\Delta\chi$
$\displaystyle=\chi-\bar{\chi}=-\frac{2\pi\mathcal{N}}{m}(f-\bar{f})=-\frac{2\pi\mathcal{N}}{m}{\Delta
f}\ ,$ (11)

where $\mathcal{N}$ is the scattering density, $m$ the kaon mass, and $f$ and
$\bar{f}$ the forward scattering amplitudes. The imaginary part of $f$ is
related to the total cross section through the optical theorem
$\sigma_{T}=(4\pi/p){\rm Im}f$. The difference in the interaction cross
sections of ${\kern 1.99997pt\overline{\kern-1.99997ptK}{}}{}^{0}$ and
${K}^{0}$ depends on the momentum of the kaon and on the number of nucleons,
$A$, in the target and is obtained from Ref. [25],

$\Delta\sigma=\sigma_{T}({{\kern
1.99997pt\overline{\kern-1.99997ptK}{}}{}^{0}})-\sigma_{T}({{K}^{0}})=23.2\,A^{0.758}[p({\mathrm{Ge\kern-1.00006ptV\\!/}c})]^{-0.614}\rm\,mb\
.$ (12)

The phase of $\Delta f$ is determined using the phase-power relation [25, 26]
to be $\arg(\Delta f)=(-124.7\pm 0.8)^{\circ}$. The regeneration incorporates
two effects [23, 27]. The term ${\rm Im}(\Delta f)$ describes the incoherent
regeneration due to absorption and elastic scattering, which is equivalent to
the case of charged kaons. The term ${\rm Re}(\Delta f)$ describes the
coherent regeneration due to dispersion (phase shift) of the ${K}^{0}$ and
${\kern 1.99997pt\overline{\kern-1.99997ptK}{}}{}^{0}$ states.

As the neutral kaons are produced in a flavour eigenstate, the initial
amplitudes at $t=0$, ${\alpha_{\rm\scriptscriptstyle L,S}}(0)$, need to be
written in the ${K}^{0}_{\rm\scriptscriptstyle L}$ and
${K}^{0}_{\rm\scriptscriptstyle S}$ basis,

$|{{K}^{0}}\rangle,|{{\kern
1.99997pt\overline{\kern-1.99997ptK}{}}{}^{0}}\rangle=\sqrt{\frac{1+|\epsilon|^{2}}{2}}\frac{1}{1\pm\epsilon}\left[|{{K}^{0}_{\rm\scriptscriptstyle
L}}\rangle\pm|{{K}^{0}_{\rm\scriptscriptstyle S}}\rangle\right]\ ,$ (13)

where $\epsilon$ describes $C\\!P$ violation in kaon mixing. At a given time,
the decay rate into the final state ${{\pi}^{+}}{{\pi}^{-}}$ is given by
$|{\alpha_{\rm\scriptscriptstyle
S}}(t)+\epsilon\,{\alpha_{\rm\scriptscriptstyle L}}(t)|^{2}$. The values of
the parameters used to calculate the ${K}^{0}$ asymmetry are given in Table 2.

Table 2: Values of the parameters used to calculate the ${K}^{0}$ asymmetry [21, 25]. Parameter | Value
---|---
$\Delta m$ | $(0.5293\pm 0.0009)\times 10^{10}\hbar\rm{\,s}^{-1}$
${\tau_{\rm\scriptscriptstyle S}}\equiv 1/{\Gamma_{\rm\scriptscriptstyle S}}$ | $(0.8954\pm 0.0004)\times 10^{-10}\rm{\,s}$
${\tau_{\rm\scriptscriptstyle L}}\equiv 1/{\Gamma_{\rm\scriptscriptstyle L}}$ | $(5.116\pm 0.021)\times 10^{-8}\rm{\,s}$
$m$ | $(497.614\pm 0.024){\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$
$\arg(\Delta f)$ | $(-124.7\pm 0.8)^{\circ}$
$|\epsilon|$ | $(2.228\pm 0.011)\times 10^{-3}$
$\phi_{+-}\equiv\arg\epsilon$ | $(43.51\pm 0.05)^{\circ}$

Using the ${K}^{0}_{\rm\scriptscriptstyle S}$ and ${D}^{0}$ decay positions,
the path of the ${K}^{0}_{\rm\scriptscriptstyle S}$ meson through the detector
is known and the expected ${K}^{0}$ asymmetry can be calculated using the
formulae above. For every ${{D}^{+}}\\!\rightarrow{{\kern
1.99997pt\overline{\kern-1.99997ptK}{}}{}^{0}}{{\pi}^{+}}$ candidate, the path
of the neutral kaon is divided into small steps using the material model of
the LHCb detector. At each step, the amplitudes are updated using Eq. (10)
starting with either a ${K}^{0}$ or ${\kern
1.99997pt\overline{\kern-1.99997ptK}{}}{}^{0}$ as initial state. The expected
${K}^{0}$ asymmetry for a given event is then the asymmetry in the decay rates
between the ${K}^{0}$ and ${\kern
1.99997pt\overline{\kern-1.99997ptK}{}}{}^{0}$ initial states. The overall
asymmetry is calculated from the expected asymmetry averaged over all
reconstructed ${{D}^{+}}\\!\rightarrow{{\kern
1.99997pt\overline{\kern-1.99997ptK}{}}{}^{0}}{{\pi}^{+}}$ candidates.

The measured raw asymmetry in the ${{D}^{+}}\\!\rightarrow{{\kern
1.99997pt\overline{\kern-1.99997ptK}{}}{}^{0}}{{\pi}^{+}}$ decay and the
effect from the predicted ${K}^{0}$ asymmetry are shown as functions of the
${K}^{0}_{\rm\scriptscriptstyle S}$ decay time in Fig. 2. The measured raw
asymmetry receives contributions not only from the ${K}^{0}$ detection
asymmetry, but also from the pion tracking asymmetry and ${D}^{+}$ production
asymmetry. These contributions are almost independent of the
${K}^{0}_{\rm\scriptscriptstyle S}$ decay time. Therefore, an overall shift is
applied to the predicted asymmetry to match the data. Assuming a negligible
pion detection asymmetry, this shift agrees well with the ${D}^{+}$ production
asymmetry of $(-0.96\pm 0.26)\%$ measured on 2011 data [28]. The downward
trend coming from the ${K}^{0}$ asymmetry is clearly visible, in particular
for downstream ${K}^{0}_{\rm\scriptscriptstyle S}$ decays. The predicted
asymmetry dependence agrees well with the data, with $p$-values of 0.81 and
0.31, respectively.

(a)Long ${K}^{0}_{\rm\scriptscriptstyle S}$(b)Downstream
${K}^{0}_{\rm\scriptscriptstyle S}$

Figure 2: Raw asymmetry in the ${{D}^{+}}\\!\rightarrow{{\kern
1.79997pt\overline{\kern-1.79997ptK}{}}{}^{0}}{{\pi}^{+}}$ decay shown for (a)
long and (b) downstream ${K}^{0}_{\rm\scriptscriptstyle S}$ candidates versus
the ${K}^{0}_{\rm\scriptscriptstyle S}$ decay time in units of its lifetime.
The long ${K}^{0}_{\rm\scriptscriptstyle S}$ candidates are reconstructed in
the full data set, while the downstream ${K}^{0}_{\rm\scriptscriptstyle S}$
candidates are reconstructed in the 2012 data only. The predicted effect from
the ${K}^{0}$ asymmetry, $-A_{D}({{K}^{0}})$, is also shown. An overall shift
is applied to this prediction to account for ${D}^{+}$ production and pion
detection asymmetries (note that the unshifted $A_{D}({{K}^{0}})$ at $t=0$ is
zero).

Only ${K}^{0}_{\rm\scriptscriptstyle S}$ candidates reconstructed with long
tracks are used in the $A_{{C\\!P}}({{K}^{-}}{{K}^{+}})$ measurement. These
candidates probe lower ${K}^{0}_{\rm\scriptscriptstyle S}$ decay times,
compared to those reconstructed with downstream tracks, resulting in a much
smaller ${K}^{0}$ asymmetry correction. Nevertheless, the effect observed in
downstream ${K}^{0}_{\rm\scriptscriptstyle S}$ decays is used to test the
accuracy of the ${K}^{0}$ detection asymmetry model. The measured difference
in raw asymmetry between the samples is $(0.49\pm 0.12)\%$, which is obtained
after weighting the long ${K}^{0}_{\rm\scriptscriptstyle S}$ sample to correct
for differences in the ${D}^{+}$ production and pion detection asymmetries.
This value agrees with the expected difference of $(0.546\pm 0.027)\%$, where
the uncertainty is dominated by the uncertainty on the amount of detector
material. The relative uncertainty of the measured difference ($25\%$) is
assigned as a systematic uncertainty on the ${K}^{0}$ asymmetry model. The
expected ${K}^{0}$ asymmetry in the ${{D}^{+}}\\!\rightarrow{{\kern
1.99997pt\overline{\kern-1.99997ptK}{}}{}^{0}}{{\pi}^{+}}$ sample with long
${K}^{0}_{\rm\scriptscriptstyle S}$ candidates weighted according to the
procedure described in Sect. 5.2 is found to be $A_{D}({{K}^{0}})=(0.054\pm
0.014\mathrm{\,(syst)})\%$.

###  5.4 Wrong flavour tags

If a ${D}^{0}$ meson is combined with a muon that does not originate from the
corresponding semileptonic $B$ decay, the ${D}^{0}$ flavour may not be
correctly assigned. The probability to wrongly tag a ${D}^{0}$ meson is
denoted by $\omega$. This mistag probability dilutes the observed asymmetry by
a factor $1-2\omega$. For small $\omega$, the expression of $\Delta
A_{{C\\!P}}$ can be written as

$\Delta A_{{C\\!P}}=(1+2\omega)[A_{\rm raw}({{K}^{-}}{{K}^{+}})-A_{\rm
raw}({{\pi}^{-}}{{\pi}^{+}})]\ .$ (14)

The mistag probability only affects the semileptonic decay modes as the
flavour of the ${D}^{+}$ reconstruction is unambiguous. In the reconstruction
of the ${{D}^{0}}\\!\rightarrow{{K}^{-}}{{\pi}^{+}}$ decay, wrong-sign decays
coming from doubly Cabibbo-suppressed
${{D}^{0}}\rightarrow{{K}^{+}}{{\pi}^{-}}$ decays and mixed
${{D}^{0}}\rightarrow{{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}\\!\rightarrow{{K}^{+}}{{\pi}^{-}}$
decays are included. Hence, the calculation of the $C\\!P$ asymmetry of the
${{D}^{0}}\\!\rightarrow{{K}^{-}}{{K}^{+}}$ decay is

$A_{{C\\!P}}({{K}^{-}}{{K}^{+}})=(1+2\omega)[A_{\rm
raw}({{K}^{-}}{{K}^{+}})-A_{\rm
raw}({{K}^{-}}{{\pi}^{+}})]+(1-2R)A_{D}({{K}^{-}}{{\pi}^{+}})\ ,$ (15)

where $R$ is the ratio of branching fractions of wrong-sign
${{D}^{0}}\rightarrow{{K}^{+}}{{\pi}^{-}}$ decays over right-sign
${{D}^{0}}\\!\rightarrow{{K}^{-}}{{\pi}^{+}}$ decays.

The ${{D}^{0}}\\!\rightarrow{{K}^{-}}{{\pi}^{+}}$ sample from semileptonic $B$
decays is also used to estimate the mistag probability. The final state,
either ${{K}^{+}}{{\pi}^{-}}$ or ${{K}^{-}}{{\pi}^{+}}$, almost unambiguously
determines the flavour of the ${D}^{0}$ meson, since the contamination from
wrong-sign decays is only $R=(0.389\pm 0.003)\%$ [29]. After correcting for
this wrong-sign fraction, the mistag probability is found to be
$\omega=(0.988\pm 0.006)\%$ for the $\Delta A_{{C\\!P}}$ measurement and
$\omega=(0.791\pm 0.006)\%$ for the $A_{{C\\!P}}({{K}^{-}}{{K}^{+}})$
measurement. The small difference between these numbers is due to more
stringent trigger criteria in the latter, resulting in different kinematic
distributions for the two selections. As a consistency check, the mistag
probability is obtained in all three semileptonic samples by searching for an
additional pion from a ${{D}^{*+}}\rightarrow{{D}^{0}}{{\pi}^{+}}$ decay and
comparing the charge of this pion with that of the muon. The mistag
probabilities are found to be in good agreement with an average value of
$\omega=(0.985\pm 0.017)\%$ for the $\Delta A_{{C\\!P}}$ measurement and of
$\omega=(0.803\pm 0.019)\%$ for the $A_{{C\\!P}}({{K}^{-}}{{K}^{+}})$
measurement. Since the dilution effect from such a small $\omega$ value
results in tiny corrections to $\Delta A_{{C\\!P}}$ and
$A_{{C\\!P}}({{K}^{-}}{{K}^{+}})$, the uncertainty in this number is
neglected. A small difference of $\Delta\omega=(0.028\pm 0.011)\%$ is observed
between the probabilities to wrongly tag ${D}^{0}$ and ${\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}$ mesons. Although any non-zero
value is expected to cancel in the calculation of $\Delta A_{{C\\!P}}$ and
$A_{{C\\!P}}({{K}^{-}}{{K}^{+}})$, the full difference is conservatively taken
as a systematic uncertainty.

###  5.5 Average decay times

The time-integrated $C\\!P$ asymmetry has contributions from direct and
indirect $C\\!P$ violation, depending on the average decay time, $\left\langle
t\right\rangle$, of the ${D}^{0}$ mesons in the sample as [2]

$A_{{C\\!P}}\approx a_{{C\\!P}}^{\rm dir}-A_{\Gamma}\frac{\left\langle
t\right\rangle}{\tau}\ ,$ (16)

where $a_{{C\\!P}}^{\rm dir}$ is the direct $C\\!P$ violation term, $\tau$ the
${D}^{0}$ lifetime, and $A_{\Gamma}$ a measure of indirect $C\\!P$ violation.
The world-average value of $A_{\Gamma}$ in singly Cabibbo-suppressed ${D}^{0}$
decays is $(-0.014\pm 0.052)\%$ [8]. Assuming that this quantity is the same
for ${{D}^{0}}\\!\rightarrow{{K}^{-}}{{K}^{+}}$ and
${{D}^{0}}\\!\rightarrow{{\pi}^{-}}{{\pi}^{+}}$ decays, the sensitivity of
$\Delta A_{{C\\!P}}$ to indirect $C\\!P$ violation is introduced by the
difference in the average ${D}^{0}$ decay times between the two decay modes. A
complete discussion is given in the previous publication [11] and the same
procedure is adopted here. In this procedure, the average decay time of the
signal in each sample is determined by subtracting the decay time
distributions of background events using the sPlot technique [30], and by
correcting for decay time resolution effects. For the $\Delta A_{{C\\!P}}$
measurement, the average decay times are found to be

$\displaystyle\left\langle t\right\rangle/\tau({{K}^{-}}{{K}^{+}})$
$\displaystyle=1.082\pm 0.001\mathrm{\,(stat)}\pm 0.004\mathrm{\,(syst)}\ ,$
$\displaystyle\left\langle t\right\rangle/\tau({{\pi}^{-}}{{\pi}^{+}})$
$\displaystyle=1.068\pm 0.001\mathrm{\,(stat)}\pm 0.004\mathrm{\,(syst)}\ .$

The small difference between these numbers ($0.014\pm 0.004$) implies that
$\Delta A_{{C\\!P}}=\Delta a_{{C\\!P}}^{\rm dir}$ is an excellent
approximation. For the $A_{{C\\!P}}({{K}^{-}}{{K}^{+}})$ measurement, the
average decay time is found to be

$\left\langle t\right\rangle/\tau({{K}^{-}}{{K}^{+}})=1.051\pm
0.001\mathrm{\,(stat)}\pm 0.004\mathrm{\,(syst)}\ .$

###  5.6 $C\\!P$ asymmetry measurements

The raw asymmetries are determined with likelihood fits to the binned
${D}^{0}$ and ${D}^{+}$ mass distributions using the mass models and event
weights as described in Sects. 5.1 and 5.2. The fits are done separately for
the 2011 and 2012 data sets and for the two magnet polarities. For each data
set the mean value of the raw asymmetry is the arithmetic average of the fit
results for the two magnet polarities. The final raw asymmetry is then the
statistically weighted average over the full data set. The derivation of the
${{K}^{-}}{{\pi}^{+}}$ detection asymmetry using prompt
${{D}^{+}}\\!\rightarrow{{K}^{-}}{{\pi}^{+}}{{\pi}^{+}}$ and
${{D}^{+}}\\!\rightarrow{{\kern
1.99997pt\overline{\kern-1.99997ptK}{}}{}^{0}}{{\pi}^{+}}$ decays is shown in
Table 3. The measured asymmetry, $A_{D}({{K}^{-}}{{\pi}^{+}})=(-1.17\pm
0.12)\%$, is dominated by the different interaction cross sections of
${K}^{-}$ and ${K}^{+}$ mesons in matter. Figure 3 shows the detection
asymmetry as a function of the kaon momentum. As expected, the kaon
interaction asymmetry decreases with kaon momentum.

Table 3: Asymmetries (in %) entering the calculation of the ${{K}^{-}}{{\pi}^{+}}$ detection asymmetry for the two magnet polarities, and for the mean value. The correction for the ${K}^{0}$ asymmetry is applied in the bottom row. The mean values in the last column are obtained first by taking the arithmetic average over the magnet polarities and then by taking the weighted averages of the 2011 and 2012 data sets. The uncertainties are statistical only. | Magnet up | Magnet down | Mean
---|---|---|---
$A_{\rm raw}({{K}^{-}}{{\pi}^{+}}{{\pi}^{+}})$ |  $-1.969$$\pm$ | $0.033$ |  $-1.672$$\pm$ | $0.032$ |  $-1.827$$\pm$ | $0.023$
$A_{\rm raw}({{\kern 1.99997pt\overline{\kern-1.99997ptK}{}}{}^{0}}{{\pi}^{+}})$ |  $-0.94$$\pm$ | $0.17$ |  $-0.51$$\pm$ | $0.16$ |  $-0.71$$\pm$ | $0.12$
$A_{D}({{K}^{-}}{{\pi}^{+}})$ |  $-1.08$$\pm$ | $0.17$ |  $-1.22$$\pm$ | $0.16$ |  $-1.17$$\pm$ | $0.12$
Figure 3: Measured ${{K}^{-}}{{\pi}^{+}}$ detection asymmetry as a function of
the kaon momentum. The shaded band indicates the average asymmetry integrated
over the bins. There is a correlation between the data points due to the
overlap between the ${{D}^{+}}\\!\rightarrow{{\kern
1.99997pt\overline{\kern-1.99997ptK}{}}{}^{0}}{{\pi}^{+}}$ samples used for
each bin.

For illustration, Fig. 4 shows the raw asymmetries for
${{D}^{0}}\\!\rightarrow{{K}^{-}}{{K}^{+}}$ and
${{D}^{0}}\\!\rightarrow{{\pi}^{-}}{{\pi}^{+}}$ candidates as functions of the
invariant mass. The raw asymmetry in both decay modes is slightly negative.
The derivation of $\Delta A_{{C\\!P}}$ and $A_{{C\\!P}}({{K}^{-}}{{K}^{+}})$
from the raw asymmetries are shown in Tables 4 and 5. There is a statistical
correlation $\rho=0.23$ between the values of $\Delta A_{{C\\!P}}$ and
$A_{{C\\!P}}({{K}^{-}}{{K}^{+}})$ as they both use candidates in the
${{D}^{0}}\\!\rightarrow{{K}^{-}}{{K}^{+}}$ sample.

(a)(b)

Figure 4: Raw asymmetry, without background subtraction, as a function of the invariant mass for (a) the ${{D}^{0}}\\!\rightarrow{{K}^{-}}{{K}^{+}}$ candidates and (b) the ${{D}^{0}}\\!\rightarrow{{\pi}^{-}}{{\pi}^{+}}$ candidates for the $\Delta A_{{C\\!P}}$ selection. The result from the fit is overlaid. Table 4: Asymmetries (in %) used in the calculation of $\Delta A_{{C\\!P}}$ for the two magnet polarities. The values for $\Delta A_{{C\\!P}}$ are corrected for the mistag probability. The mean values in the last column are obtained first by taking the arithmetic average over the magnet polarities and then by taking the weighted averages of the 2011 and 2012 data sets. The uncertainties are statistical only. | Magnet up | Magnet down | Mean
---|---|---|---
$A_{\rm raw}({{K}^{-}}{{K}^{+}})$ |  $-0.46$$\pm$ | $0.11$ |  $-0.43$$\pm$ | $0.11$ |  $-0.44$$\pm$ | $0.08$
$A_{\rm raw}({{\pi}^{-}}{{\pi}^{+}})$ |  $-0.45$$\pm$ | $0.20$ |  $-0.66$$\pm$ | $0.19$ |  $-0.58$$\pm$ | $0.14$
$\Delta A_{{C\\!P}}$ |  $-0.01$$\pm$ | $0.23$ |  $+0.24$$\pm$ | $0.22$ |  $+0.14$$\pm$ | $0.16$
Table 5: Asymmetries (in %) used in the calculation of $A_{{C\\!P}}({{K}^{-}}{{K}^{+}})$ for the two magnet polarities. The values for $A_{{C\\!P}}({{K}^{-}}{{K}^{+}})$ are corrected for the mistag probability. The mean values in the last column are obtained first by taking the arithmetic average over the magnet polarities and then by taking the weighted averages of the 2011 and 2012 data sets. The uncertainties are statistical only. | Magnet up | Magnet down | Mean
---|---|---|---
$A_{\rm raw}({{K}^{-}}{{K}^{+}})$ |  $-0.45$$\pm$ | $0.12$ |  $-0.41$$\pm$ | $0.12$ |  $-0.43$$\pm$ | $0.08$
$A_{\rm raw}({{K}^{-}}{{\pi}^{+}})$ |  $-1.41$$\pm$ | $0.05$ |  $-1.59$$\pm$ | $0.05$ |  $-1.51$$\pm$ | $0.04$
$A_{D}({{K}^{-}}{{\pi}^{+}})$ |  $-1.08$$\pm$ | $0.17$ |  $-1.22$$\pm$ | $0.16$ |  $-1.17$$\pm$ | $0.12$
$A_{{C\\!P}}({{K}^{-}}{{K}^{+}})$ |  $-0.09$$\pm$ | $0.21$ |  $-0.01$$\pm$ | $0.21$ |  $-0.06$$\pm$ | $0.15$

##  6 Systematic uncertainties

Systematic shifts in the observed $C\\!P$ asymmetries can arise from non-
cancellation of production and detection asymmetries, misreconstruction of the
final state, and imperfect modelling of the background. The contributions to
the systematic uncertainties in $\Delta A_{{C\\!P}}$ and
$A_{{C\\!P}}({{K}^{-}}{{K}^{+}})$ are described below.

The fractions of ${B}^{0}$ and ${{B}^{+}}$ decays in the three semileptonic
$B$ samples can be slightly different. Assuming that there is a difference in
the ${B}^{0}$ and ${{B}^{+}}$ production asymmetries, a residual production
asymmetry can remain in $\Delta A_{{C\\!P}}$ and
$A_{{C\\!P}}({{K}^{-}}{{K}^{+}})$. As in the previous publication [11], a
systematic uncertainty of $0.02\%$ is assigned to both $\Delta A_{{C\\!P}}$
and $A_{{C\\!P}}({{K}^{-}}{{K}^{+}})$. Due to ${B}^{0}$ oscillations, the
observed $B$ production asymmetry depends on the decay-time acceptance of the
reconstructed $B$ meson, which is slightly different for the three decay
modes. This produces a systematic uncertainty of $0.02\%$ for both $\Delta
A_{{C\\!P}}$ and $A_{{C\\!P}}({{K}^{-}}{{K}^{+}})$, similar to the one found
previously [11].

The weighting procedure almost equalises the $p_{\rm T}$ and $\eta$
distributions of the particles, but small differences remain. When also
weighting for different azimuthal angle distributions of the final state
particles, the change in both $\Delta A_{{C\\!P}}$ and
$A_{{C\\!P}}({{K}^{-}}{{K}^{+}})$ is negligible. Slightly larger shifts are
seen when increasing (decreasing) the number of bins used in each kinematic
variable from 20 to 25 (15) or when changing the ${D}^{0}$ mass range. The
maximum shift, $0.02\%$ for $\Delta A_{{C\\!P}}$ and $0.05\%$ for
$A_{{C\\!P}}({{K}^{-}}{{K}^{+}})$, is taken as a systematic uncertainty.
Finally, the cancellation of the production and detection asymmetries is
tested by randomly assigning the charge of the muon or charged $D$ meson in
real data, depending on the $p_{\rm T}$ of the particles. The $B$ and
${D}^{+}$ production asymmetries and the $\mu^{\pm}$, ${K}^{\pm}$ and
${\pi}^{\pm}$ detection asymmetries that are simulated in this way are
motivated by the small $p_{\rm T}$-dependences observed in data. No shift is
seen in the value of $\Delta A_{{C\\!P}}$, while a small shift of $0.03\%$ is
observed in the value of $A_{{C\\!P}}({{K}^{-}}{{K}^{+}})$. This shift is
propagated as part of the uncertainty due the weighting procedure.

The sensitivity of the results to the signal and background models is
determined by varying the signal and background functions. The alternative
signal functions are a Johnson $S_{U}$ distribution [31], a single Gaussian,
and a double Gaussian function. The alternative background function is a
second-order polynomial. Furthermore, the effect of using different mass
binning and fit range, and the effect of constraining the asymmetry of the
${{D}^{0}}\\!\rightarrow{{K}^{-}}{{\pi}^{+}}$ background in the
${{D}^{0}}\\!\rightarrow{{\pi}^{-}}{{\pi}^{+}}$ decay to the observed
asymmetry, are considered. The maximum variations from the default fit for
each decay mode are added in quadrature to determine the systematic
uncertainty for $\Delta A_{{C\\!P}}$ ($0.06\%$) and for
$A_{{C\\!P}}({{K}^{-}}{{K}^{+}})$ ($0.06\%$).

In the default fit, the background can vary freely with an overall asymmetry
and different slope parameters for each tag. Nevertheless, background
contributions from different origins can have different shapes and
asymmetries. Such an effect is expected to be largest in the
${{D}^{0}}\\!\rightarrow{{K}^{-}}{{K}^{+}}$ decay, due to possible
contributions from other charm decays, and is studied by generating
pseudoexperiments with different background compositions in the two Cabibbo-
suppressed decays. Three types of background shapes are simulated: an
exponential function to describe the combinatorial background observed in
data, another exponential function with a different slope inspired by
partially-reconstructed background from simulated
${{D}^{+}}\\!\rightarrow{{K}^{-}}{{\pi}^{+}}{{\pi}^{+}}$ decays, and a linear
shape inspired by partially-reconstructed background from simulated
${{\mathchar 28931\relax}^{+}_{c}}\rightarrow p{{K}^{-}}{{\pi}^{+}}$ decays.
The asymmetries in the background are varied by up to $\pm 3\%$. Such large
asymmetries are incompatible with the asymmetries observed in the background
and therefore constitute an upper bound on the magnitude of any possible
effect. The largest bias in the raw asymmetry ($0.03\%$) is propagated as a
systematic uncertainty for $\Delta A_{{C\\!P}}$ and
$A_{{C\\!P}}({{K}^{-}}{{K}^{+}})$.

The systematic shift in the raw asymmetries when removing multiple candidates
is below $0.005\%$ and therefore neglected. Higher-order corrections to Eq.
(2) are at the $10^{-6}$ level and are neglected as well. The systematic
uncertainty from the neutral kaon asymmetry ($0.01\%$) is taken from Sect. 5.3
and the systematic uncertainty from wrong combinations of muons and ${D}^{0}$
mesons is taken from Sect. 5.4. All systematic uncertainties are summarised in
Table 6 for $\Delta A_{{C\\!P}}$ and $A_{{C\\!P}}({{K}^{-}}{{K}^{+}})$. The
correlation coefficient between the total systematic uncertainties is
$\rho=0.40$.

Table 6: Contributions to the systematic uncertainty of $\Delta A_{{C\\!P}}$ and $A_{{C\\!P}}({{K}^{-}}{{K}^{+}})$. Source of uncertainty | $\Delta A_{{C\\!P}}$ | $A_{{C\\!P}}({{K}^{-}}{{K}^{+}})$
---|---|---
Production asymmetry: |  |
Difference in $b$-hadron mixture | $0.02\%$ | $0.02\%$
Difference in $B$ decay time acceptance | $0.02\%$ | $0.02\%$
Production and detection asymmetry: |  |
Different weighting | $0.02\%$ | $0.05\%$
Non-cancellation | - | $0.03\%$
Neutral kaon asymmetry | - | $0.01\%$
Background from real ${D}^{0}$ mesons: |  |
Mistag asymmetry | $0.03\%$ | $0.03\%$
Background from fake ${D}^{0}$ mesons: |  |
${D}^{0}$ mass fit model | $0.06\%$ | $0.06\%$
Wrong background modelling | $0.03\%$ | $0.03\%$
Quadratic sum | $0.08\%$ | $0.10\%$

##  7 Consistency checks

As a consistency check, the raw asymmetries in the
${{D}^{0}}\\!\rightarrow{{K}^{-}}{{K}^{+}}$ and
${{D}^{0}}\\!\rightarrow{{\pi}^{-}}{{\pi}^{+}}$ samples and $\Delta
A_{{C\\!P}}$ are determined as functions of the impact parameter of the
${D}^{0}$ trajectory with respect to the primary vertex, the flight distance
of the $B$ candidate, the angle between the directions of the muon and the
${D}^{0}$ decay products, the muon and ${D}^{0}$ kinematic variables, the
reconstructed ${{D}^{0}}\mu$ invariant mass, the multiplicity of tracks and
primary vertices in the event, the particle identification requirement on the
kaons, and the selected trigger lines. No significant dependence is observed
on any of these variables. Another test is made by including ${D}^{0}$
candidates with negative decay times. In particular in the
${{D}^{0}}\\!\rightarrow{{\pi}^{-}}{{\pi}^{+}}$ decay, there is more
background at low ${D}^{0}$ decay times. Enhancing this type of background by
including negative decay time candidates does not change the values for
$\Delta A_{{C\\!P}}$ or $A_{{C\\!P}}({{K}^{-}}{{K}^{+}})$.

During periods without data taking, interventions on the detector and on the
trigger change the alignment and data acquisition conditions. This could
induce time-varying detection asymmetries. The final results should not be
sensitive to such variations as data are calibrated with control samples
collected in the same data-taking period. Nevertheless, any residual detector
asymmetry would manifest itself as variations in time of the measured $C\\!P$
asymmetries. Figure 5 shows $\Delta A_{{C\\!P}}$ and
$A_{{C\\!P}}({{K}^{-}}{{K}^{+}})$ versus data taking period. These periods are
separated by interruptions in data taking, and within each period the magnetic
field is reversed at least once. No dependence of the obtained $C\\!P$
asymmetries on the data taking period is observed. The average values of
$\Delta A_{{C\\!P}}$ are $(+0.33\pm 0.30\mathrm{\,(stat)})\%$ for the 2011
data and $(+0.06\pm 0.19\mathrm{\,(stat)})\%$ for the 2012 data. The value for
the 2011 data is slightly lower compared to the previous analysis, which is
attributed to the non-overlapping data samples, due to differences in the
selection and in the calibration of the detector. The main shift is due to the
looser particle identification requirements on the kaons in this analysis.
Such a shift is not seen in the 2012 data. The average values of
$A_{{C\\!P}}({{K}^{-}}{{K}^{+}})$ are $(+0.04\pm 0.28\mathrm{\,(stat)})\%$ for
the 2011 data and $(-0.10\pm 0.18\mathrm{\,(stat)})\%$ for the 2012 data.

(a)(b)

Figure 5: (a) $\Delta A_{{C\\!P}}$ and (b) $A_{{C\\!P}}({{K}^{-}}{{K}^{+}})$
as a function of the data taking period. The 2011 data are divided into four
periods and the 2012 data into six periods. The error bars indicate the
statistical uncertainty, the shaded bands show the averages for 2011 and 2012,
and the (red) line shows the overall $C\\!P$ asymmetry.

##  8 Conclusion

The difference in $C\\!P$ asymmetries between the
${{D}^{0}}\\!\rightarrow{{K}^{-}}{{K}^{+}}$ and
${{D}^{0}}\\!\rightarrow{{\pi}^{-}}{{\pi}^{+}}$ decay channels and the $C\\!P$
asymmetry in the ${{D}^{0}}\\!\rightarrow{{K}^{-}}{{K}^{+}}$ channel are
measured using muon-tagged ${D}^{0}$ decays in the 3$\mbox{\,fb}^{-1}$ data
set to be

$\displaystyle\Delta A_{{C\\!P}}$ $\displaystyle=(+0.14\pm
0.16\mathrm{\,(stat)}\pm 0.08\mathrm{\,(syst)})\%\ ,$ $\displaystyle
A_{{C\\!P}}({{K}^{-}}{{K}^{+}})$ $\displaystyle=(-0.06\pm
0.15\mathrm{\,(stat)}\pm 0.10\mathrm{\,(syst)})\%\ ,$

where the total correlation coefficient, including statistical and systematic
components, is $\rho=0.28$. By combining the above results, the $C\\!P$
asymmetry in the ${{D}^{0}}\\!\rightarrow{{\pi}^{-}}{{\pi}^{+}}$ decay is
found to be

$A_{{C\\!P}}({{\pi}^{-}}{{\pi}^{+}})=(-0.20\pm 0.19\mathrm{\,(stat)}\pm
0.10\mathrm{\,(syst)})\%\ .$

These results are obtained assuming that there is no $C\\!P$ violation in
${D}^{0}$ mixing and no direct $C\\!P$ violation in the Cabibbo-favoured
${{D}^{0}}\\!\rightarrow{{K}^{-}}{{\pi}^{+}}$,
${{D}^{+}}\\!\rightarrow{{K}^{-}}{{\pi}^{+}}{{\pi}^{+}}$ and
${{D}^{+}}\\!\rightarrow{{\kern
1.99997pt\overline{\kern-1.99997ptK}{}}{}^{0}}{{\pi}^{+}}$ decay modes. The
measurement of $\Delta A_{{C\\!P}}$ supersedes the previously reported result
[11]. Our results show that there is no significant $C\\!P$ violation in the
singly Cabibbo-suppressed
${{D}^{0}}\rightarrow{{K}^{-}}{{K}^{+}},{{\pi}^{-}}{{\pi}^{+}}$ decays at the
level of $10^{-3}$. These results constitute the most precise measurements of
time-integrated $C\\!P$ asymmetries $A_{{C\\!P}}({{K}^{-}}{{K}^{+}})$ and
$A_{{C\\!P}}({{\pi}^{-}}{{\pi}^{+}})$ from a single experiment to date.

## 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); MEN/IFA (Romania); MinES, Rosatom, RFBR and NRC “Kurchatov
Institute” (Russia); MinECo, XuntaGal and GENCAT (Spain); SNSF and SER
(Switzerland); NASU (Ukraine); STFC and the Royal Society (United Kingdom);
NSF (USA). We also acknowledge the support received from EPLANET, Marie Curie
Actions and the ERC under FP7. The Tier1 computing centres are supported by
IN2P3 (France), KIT and BMBF (Germany), INFN (Italy), NWO and SURF (The
Netherlands), PIC (Spain), GridPP (United Kingdom). We are indebted to the
communities behind the multiple open source software packages on which we
depend. We are also thankful for the computing resources and the access to
software R&D tools provided by Yandex LLC (Russia).

## References

  * [1] M. Bobrowski, A. Lenz, J. Riedl, and J. Rohrwild, How large can the SM contribution to CP Violation in ${{D}^{0}}-{{\kern 1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}$ mixing be?, JHEP 03 (2010) 009, arXiv:1002.4794
  * [2] Y. Grossman, A. L. Kagan, and Y. Nir, New physics and CP violation in singly Cabibbo suppressed D decays, Phys. Rev. D75 (2007) 036008, arXiv:hep-ph/0609178
  * [3] T. Feldmann, S. Nandi, and A. Soni, Repercussions of flavour symmetry breaking on CP violation in D-meson decays, JHEP 06 (2012) 007, arXiv:1202.3795
  * [4] J. Brod, A. L. Kagan, and J. Zupan, Size of direct CP violation in singly Cabibbo-suppressed D decays, Phys. Rev. D86 (2012) 014023, arXiv:1111.5000
  * [5] B. Bhattacharya, M. Gronau, and J. L. Rosner, CP asymmetries in singly-Cabibbo-suppressed $D$ decays to two pseudoscalar mesons, Phys. Rev. D85 (2012) 054014, arXiv:1201.2351
  * [6] LHCb collaboration, R. Aaij, _et al._ , and A. Bharucha et al., Implications of LHCb measurements and future prospects, Eur. Phys. J. C73 (2013) 2373, arXiv:1208.3355
  * [7] S. Bianco, F. Fabbri, D. Benson, and I. Bigi, A Cicerone for the physics of charm, Riv. Nuovo Cim. 26N7 (2003) 1, arXiv:hep-ex/0309021
  * [8] Heavy Flavor Averaging Group, Y. Amhis et al., Averages of b-hadron, c-hadron, and tau-lepton properties as of early 2012, arXiv:1207.1158, updated averages and plots available from http://www.slac.stanford.edu/xorg/hfag/
  * [9] LHCb collaboration, R. Aaij et al., Evidence for $C\\!P$ violation in time-integrated $D^{0}\rightarrow h^{-}h^{+}$ decay rates, Phys. Rev. Lett. 108 (2012) 111602, arXiv:1112.0938
  * [10] Superseded by LHCb collaboration, A search for time-integrated $C\\!P$ violation in $D^{0}\rightarrow K^{-}K^{+}$ and $D^{0}\rightarrow\pi^{-}\pi^{+}$ decays, LHCb-CONF-2013-003
  * [11] LHCb collaboration, R. Aaij et al., Search for direct $C\\!P$ violation in $D^{0}\rightarrow h^{-}h^{+}$ modes using semileptonic $B$ decays, Phys. Lett. B723 (2013) 33, arXiv:1303.2614
  * [12] BaBar collaboration, B. Aubert et al., Search for CP violation in the decays $D^{0}\rightarrow K^{-}K^{+}$ and $D^{0}\rightarrow\pi^{-}\pi^{+}$, Phys. Rev. Lett. 100 (2008) 061803, arXiv:0709.2715
  * [13] CDF collaboration, T. Aaltonen et al., Measurement of CP-violating asymmetries in $D^{0}\rightarrow\pi^{+}\pi^{-}$ and $D^{0}\rightarrow K^{+}K^{-}$ decays at CDF, Phys. Rev. D85 (2012) 012009, arXiv:1111.5023
  * [14] CDF collaboration, T. Aaltonen et al., Measurement of the difference of CP–violating asymmetries in $D^{0}\rightarrow K^{+}K^{-}$ and $D^{0}\rightarrow\pi^{+}\pi^{-}$ decays at CDF, Phys. Rev. Lett. 109 (2012) 111801, arXiv:1207.2158, combined average available from http://www-cdf.fnal.gov/physics/new/bottom/120216.blessed-CPVcharm10fb/
  * [15] Belle collaboration, M. Staric et al., Measurement of CP asymmetry in Cabibbo suppressed $D^{0}$ decays, Phys. Lett. B670 (2008) 190, arXiv:0807.0148
  * [16] Belle collaboration, B. Ko, Direct CP violation in charm at Belle, PoS ICHEP2012 (2013) 353, arXiv:1212.1975
  * [17] LHCb collaboration, A. A. Alves Jr. et al., The LHCb detector at the LHC, JINST 3 (2008) S08005
  * [18] M. Adinolfi et al., Performance of the LHCb RICH detector at the LHC, Eur. Phys. J. C73 (2013) 2431, arXiv:1211.6759
  * [19] R. Aaij et al., The LHCb trigger and its performance in 2011, JINST 8 (2013) P04022, arXiv:1211.3055
  * [20] W. D. Hulsbergen, Decay chain fitting with a Kalman filter, Nucl. Instrum. Meth. A552 (2005) 566, arXiv:physics/0503191
  * [21] Particle Data Group, J. Beringer et al., Review of particle physics, Phys. Rev. D86 (2012) 010001, and 2013 partial update for the 2014 edition
  * [22] LHCb collaboration, R. Aaij et al., Precision measurements of $D$ meson mass differences, JHEP 06 (2013) 65, arXiv:1304.6865
  * [23] A. Pais and O. Piccioni, Note on the decay and absorption of the $\theta^{0}$, Phys. Rev. 100 (1955) 1487
  * [24] W. Fetscher et al., Regeneration of arbitrary coherent neutral kaon states: A new method for measuring the ${{K}^{0}}-{{\kern 1.99997pt\overline{\kern-1.99997ptK}{}}{}^{0}}$ forward scattering amplitude, Z. Phys. C72 (1996) 543
  * [25] A. Gsponer et al., Precise coherent ${K}^{0}_{\rm\scriptscriptstyle S}$ regeneration amplitudes for C, Al, Cu, Sn and Pb nuclei from 20 to $140{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ and their interpretation, Phys. Rev. Lett. 42 (1979) 13
  * [26] R. A. Briere and B. Winstein, Determining the phase of a strong scattering amplitude from its momentum dependence to better than 1∘: The example of kaon regeneration, Phys. Rev. Lett. 75 (1995) 402
  * [27] M. L. Good, Relation between scattering and absorption in the Pais-Piccioni phenomenon, Phys. Rev. 106 (1957) 591
  * [28] LHCb collaboration, R. Aaij et al., Measurement of the $D^{\pm}$ production asymmetry in $7~{}T\kern-0.50003pteV$ $pp$ collisions, Phys. Lett. B718 (2013) 902–909, arXiv:1210.4112
  * [29] LHCb collaboration, R. Aaij et al., Measurement of ${{D}^{0}}$–${{\kern 1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}$ mixing parameters and search for $C\\!P$ violation using ${{D}^{0}}\rightarrow{{K}^{+}}{{\pi}^{-}}$ decays, Phys. Rev. Lett. 111 (2013) 251801, arXiv:1309.6534
  * [30] M. Pivk and F. R. Le Diberder, sPlot: a statistical tool to unfold data distributions, Nucl. Instrum. Meth. A555 (2005) 356, arXiv:physics/0402083
  * [31] N. Johnson, Systems of frequency curves generated by methods of translation, Biometrika 36 (1949) 149