EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)

LHCb-PAPER-2011-017CERN-PH-EP-2011-163

December 26, 2013

Search for CP violation in D+ → K−K+π+ decays

R. Aaij23, B. Adeva36, M. Adinolfi42, C. Adrover6, A. Affolder48, Z. Ajaltouni5, J. Albrecht37, F. Alessio37,

M. Alexander47, G. Alkhazov29, P. Alvarez Cartelle36, A.A. Alves Jr22, S. Amato2, Y. Amhis38, J. Anderson39,

R.B. Appleby50, O. Aquines Gutierrez10, F. Archilli18,37, L. Arrabito53, A. Artamonov 34, M. Artuso52,37,

E. Aslanides6, G. Auriemma22,m, S. Bachmann11, J.J. Back44, D.S. Bailey50, V. Balagura30,37, W. Baldini16,

R.J. Barlow50, C. Barschel37, S. Barsuk7, W. Barter43, A. Bates47, C. Bauer10, Th. Bauer23, A. Bay38, I. Bediaga1,

K. Belous34, I. Belyaev30,37, E. Ben-Haim8, M. Benayoun8, G. Bencivenni18, S. Benson46, J. Benton42, R. Bernet39,

M.-O. Bettler17, M. van Beuzekom23, A. Bien11, S. Bifani12, A. Bizzeti17,h, P.M. Bjørnstad50, T. Blake49,

F. Blanc38, C. Blanks49, J. Blouw11, S. Blusk52, A. Bobrov33, V. Bocci22, A. Bondar33, N. Bondar29,

W. Bonivento15, S. Borghi47, A. Borgia52, T.J.V. Bowcock48, C. Bozzi16, T. Brambach9, J. van den Brand24,

J. Bressieux38, D. Brett50, S. Brisbane51, M. Britsch10, T. Britton52, N.H. Brook42, H. Brown48,

A. Buchler-Germann39, I. Burducea28, A. Bursche39, J. Buytaert37, S. Cadeddu15, J.M. Caicedo Carvajal37,

O. Callot7, M. Calvi20,j , M. Calvo Gomez35,n, A. Camboni35, P. Campana18,37, A. Carbone14, G. Carboni21,k,

R. Cardinale19,i,37, A. Cardini15, L. Carson36, K. Carvalho Akiba23, G. Casse48, M. Cattaneo37, M. Charles51,

Ph. Charpentier37, N. Chiapolini39, K. Ciba37, X. Cid Vidal36, G. Ciezarek49, P.E.L. Clarke46,37, M. Clemencic37,

H.V. Cliff43, J. Closier37, C. Coca28, V. Coco23, J. Cogan6, P. Collins37, F. Constantin28, G. Conti38, A. Contu51,

A. Cook42, M. Coombes42, G. Corti37, G.A. Cowan38, R. Currie46, B. D’Almagne7, C. D’Ambrosio37, P. David8,

I. De Bonis4, S. De Capua21,k, M. De Cian39, F. De Lorenzi12, J.M. De Miranda1, L. De Paula2, P. De Simone18,

D. Decamp4, M. Deckenhoff9, H. Degaudenzi38,37, M. Deissenroth11, L. Del Buono8, C. Deplano15, O. Deschamps5,

F. Dettori15,d, J. Dickens43, H. Dijkstra37, P. Diniz Batista1, S. Donleavy48, F. Dordei11, A. Dosil Suarez36,

D. Dossett44, A. Dovbnya40, F. Dupertuis38, R. Dzhelyadin34, C. Eames49, S. Easo45, U. Egede49, V. Egorychev30,

S. Eidelman33, D. van Eijk23, F. Eisele11, S. Eisenhardt46, R. Ekelhof9, L. Eklund47, Ch. Elsasser39,

D.G. d’Enterria35,o, D. Esperante Pereira36, L. Esteve43, A. Falabella16,e, E. Fanchini20,j , C. Farber11, G. Fardell46,

C. Farinelli23, S. Farry12, V. Fave38, V. Fernandez Albor36, M. Ferro-Luzzi37, S. Filippov32, C. Fitzpatrick46,

M. Fontana10, F. Fontanelli19,i, R. Forty37, M. Frank37, C. Frei37, M. Frosini17,f,37, S. Furcas20,

A. Gallas Torreira36, D. Galli14,c, M. Gandelman2, P. Gandini51, Y. Gao3, J-C. Garnier37, J. Garofoli52,

J. Garra Tico43, L. Garrido35, C. Gaspar37, N. Gauvin38, M. Gersabeck37, T. Gershon44,37, Ph. Ghez4, V. Gibson43,

V.V. Gligorov37, C. Gobel54, D. Golubkov30, A. Golutvin49,30,37, A. Gomes2, H. Gordon51,M. Grabalosa Gandara35, R. Graciani Diaz35, L.A. Granado Cardoso37, E. Grauges35, G. Graziani17, A. Grecu28,

S. Gregson43, B. Gui52, E. Gushchin32, Yu. Guz34, T. Gys37, G. Haefeli38, C. Haen37, S.C. Haines43, T. Hampson42,

S. Hansmann-Menzemer11, R. Harji49, N. Harnew51, J. Harrison50, P.F. Harrison44, J. He7, V. Heijne23,

K. Hennessy48, P. Henrard5, J.A. Hernando Morata36, E. van Herwijnen37, E. Hicks48, W. Hofmann10,

K. Holubyev11, P. Hopchev4, W. Hulsbergen23, P. Hunt51, T. Huse48, R.S. Huston12, D. Hutchcroft48, D. Hynds47,

V. Iakovenko41, P. Ilten12, J. Imong42, R. Jacobsson37, A. Jaeger11, M. Jahjah Hussein5, E. Jans23, F. Jansen23,

P. Jaton38, B. Jean-Marie7, F. Jing3, M. John51, D. Johnson51, C.R. Jones43, B. Jost37, S. Kandybei40,

M. Karacson37, T.M. Karbach9, J. Keaveney12, U. Kerzel37, T. Ketel24, A. Keune38, B. Khanji6, Y.M. Kim46,

M. Knecht38, S. Koblitz37, P. Koppenburg23, A. Kozlinskiy23, L. Kravchuk32, K. Kreplin11, M. Kreps44,

G. Krocker11, P. Krokovny11, F. Kruse9, K. Kruzelecki37, M. Kucharczyk20,25,37, S. Kukulak25, R. Kumar14,37,

T. Kvaratskheliya30,37, V.N. La Thi38, D. Lacarrere37, G. Lafferty50, A. Lai15, D. Lambert46, R.W. Lambert37,

E. Lanciotti37, G. Lanfranchi18, C. Langenbruch11, T. Latham44, R. Le Gac6, J. van Leerdam23, J.-P. Lees4,

R. Lefevre5, A. Leflat31,37, J. Lefrancois7, O. Leroy6, T. Lesiak25, L. Li3, L. Li Gioi5, M. Lieng9, M. Liles48,

R. Lindner37, C. Linn11, B. Liu3, G. Liu37, J.H. Lopes2, E. Lopez Asamar35, N. Lopez-March38, J. Luisier38,

F. Machefert7, I.V. Machikhiliyan4,30, F. Maciuc10, O. Maev29,37, J. Magnin1, S. Malde51, R.M.D. Mamunur37,

G. Manca15,d, G. Mancinelli6, N. Mangiafave43, U. Marconi14, R. Marki38, J. Marks11, G. Martellotti22,

A. Martens7, L. Martin51, A. Martın Sanchez7, D. Martinez Santos37, D. Martins Tostes1, A. Massafferri1,

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R. McNulty12, C. Mclean14, M. Meissner11, M. Merk23, J. Merkel9, R. Messi21,k, S. Miglioranzi37, D.A. Milanes13,37,

M.-N. Minard4, J. Molina Rodriguez54, S. Monteil5, D. Moran12, P. Morawski25, R. Mountain52, I. Mous23,

F. Muheim46, K. Muller39, R. Muresan28,38, B. Muryn26, M. Musy35, J. Mylroie-Smith48, P. Naik42, T. Nakada38,

R. Nandakumar45, J. Nardulli45, I. Nasteva1, M. Nedos9, M. Needham46, N. Neufeld37, C. Nguyen-Mau38,p,

M. Nicol7, S. Nies9, V. Niess5, N. Nikitin31, A. Oblakowska-Mucha26, V. Obraztsov34, S. Oggero23, S. Ogilvy47,

O. Okhrimenko41, R. Oldeman15,d, M. Orlandea28, J.M. Otalora Goicochea2, P. Owen49, B. Pal52, J. Palacios39,

M. Palutan18, J. Panman37, A. Papanestis45, M. Pappagallo13,b, C. Parkes47,37, C.J. Parkinson49, G. Passaleva17,

G.D. Patel48, M. Patel49, S.K. Paterson49, G.N. Patrick45, C. Patrignani19,i, C. Pavel-Nicorescu28,

A. Pazos Alvarez36, A. Pellegrino23, G. Penso22,l, M. Pepe Altarelli37, S. Perazzini14,c, D.L. Perego20,j ,

E. Perez Trigo36, A. Perez-Calero Yzquierdo35, P. Perret5, M. Perrin-Terrin6, G. Pessina20, A. Petrella16,37,

A. Petrolini19,i, B. Pie Valls35, B. Pietrzyk4, T. Pilar44, D. Pinci22, R. Plackett47, S. Playfer46, M. Plo Casasus36,

G. Polok25, A. Poluektov44,33, E. Polycarpo2, D. Popov10, B. Popovici28, C. Potterat35, A. Powell51, T. du Pree23,

J. Prisciandaro38, V. Pugatch41, A. Puig Navarro35, W. Qian52, J.H. Rademacker42, B. Rakotomiaramanana38,

M.S. Rangel2, I. Raniuk40, G. Raven24, S. Redford51, M.M. Reid44, A.C. dos Reis1, S. Ricciardi45, K. Rinnert48,

D.A. Roa Romero5, P. Robbe7, E. Rodrigues47, F. Rodrigues2, P. Rodriguez Perez36, G.J. Rogers43, S. Roiser37,

V. Romanovsky34, J. Rouvinet38, T. Ruf37, H. Ruiz35, G. Sabatino21,k, J.J. Saborido Silva36, N. Sagidova29,

P. Sail47, B. Saitta15,d, C. Salzmann39, M. Sannino19,i, R. Santacesaria22, C. Santamarina Rios36, R. Santinelli37,

E. Santovetti21,k, M. Sapunov6, A. Sarti18,l, C. Satriano22,m, A. Satta21, M. Savrie16,e, D. Savrina30, P. Schaack49,

M. Schiller11, S. Schleich9, M. Schmelling10, B. Schmidt37, O. Schneider38, A. Schopper37, M.-H. Schune7,

R. Schwemmer37, A. Sciubba18,l, M. Seco36, A. Semennikov30, K. Senderowska26, I. Sepp49, N. Serra39, J. Serrano6,

P. Seyfert11, B. Shao3, M. Shapkin34, I. Shapoval40,37, P. Shatalov30, Y. Shcheglov29, T. Shears48, L. Shekhtman33,

O. Shevchenko40, V. Shevchenko30, A. Shires49, R. Silva Coutinho54, H.P. Skottowe43, T. Skwarnicki52,

A.C. Smith37, N.A. Smith48, K. Sobczak5, F.J.P. Soler47, A. Solomin42, F. Soomro49, B. Souza De Paula2,

B. Spaan9, A. Sparkes46, P. Spradlin47, F. Stagni37, S. Stahl11, O. Steinkamp39, S. Stoica28, S. Stone52,37,

B. Storaci23, M. Straticiuc28, U. Straumann39, N. Styles46, V.K. Subbiah37, S. Swientek9, M. Szczekowski27,

P. Szczypka38, T. Szumlak26, S. T’Jampens4, E. Teodorescu28, F. Teubert37, C. Thomas51,45, E. Thomas37,

J. van Tilburg11, V. Tisserand4, M. Tobin39, S. Topp-Joergensen51, M.T. Tran38, A. Tsaregorodtsev6, N. Tuning23,

M. Ubeda Garcia37, A. Ukleja27, P. Urquijo52, U. Uwer11, V. Vagnoni14, G. Valenti14, R. Vazquez Gomez35,

P. Vazquez Regueiro36, S. Vecchi16, J.J. Velthuis42, M. Veltri17,g, K. Vervink37, B. Viaud7, I. Videau7, D. Vieira2,

X. Vilasis-Cardona35,n, J. Visniakov36, A. Vollhardt39, D. Voong42, A. Vorobyev29, H. Voss10, K. Wacker9,

S. Wandernoth11, J. Wang52, D.R. Ward43, A.D. Webber50, D. Websdale49, M. Whitehead44, D. Wiedner11,

L. Wiggers23, G. Wilkinson51, M.P. Williams44,45, M. Williams49, F.F. Wilson45, J. Wishahi9, M. Witek25,37,

W. Witzeling37, S.A. Wotton43, K. Wyllie37, Y. Xie46, F. Xing51, Z. Yang3, R. Young46, O. Yushchenko34,

M. Zavertyaev10,a, F. Zhang3, L. Zhang52, W.C. Zhang12, Y. Zhang3, A. Zhelezov11, L. Zhong3, E. Zverev31,

A. Zvyagin 37.

1Centro Brasileiro de Pesquisas Fısicas (CBPF), Rio de Janeiro, Brazil2Universidade Federal do Rio de Janeiro (UFRJ), Rio de Janeiro, Brazil3Center for High Energy Physics, Tsinghua University, Beijing, China4LAPP, Universite de Savoie, CNRS/IN2P3, Annecy-Le-Vieux, France

5Clermont Universite, Universite Blaise Pascal, CNRS/IN2P3, LPC, Clermont-Ferrand, France6CPPM, Aix-Marseille Universite, CNRS/IN2P3, Marseille, France

7LAL, Universite Paris-Sud, CNRS/IN2P3, Orsay, France8LPNHE, Universite Pierre et Marie Curie, Universite Paris Diderot, CNRS/IN2P3, Paris, France

9Fakultat Physik, Technische Universitat Dortmund, Dortmund, Germany10Max-Planck-Institut fur Kernphysik (MPIK), Heidelberg, Germany

11Physikalisches Institut, Ruprecht-Karls-Universitat Heidelberg, Heidelberg, Germany12School of Physics, University College Dublin, Dublin, Ireland

13Sezione INFN di Bari, Bari, Italy14Sezione INFN di Bologna, Bologna, Italy15Sezione INFN di Cagliari, Cagliari, Italy16Sezione INFN di Ferrara, Ferrara, Italy17Sezione INFN di Firenze, Firenze, Italy

18Laboratori Nazionali dell’INFN di Frascati, Frascati, Italy19Sezione INFN di Genova, Genova, Italy

20Sezione INFN di Milano Bicocca, Milano, Italy21Sezione INFN di Roma Tor Vergata, Roma, Italy

3

22Sezione INFN di Roma La Sapienza, Roma, Italy23Nikhef National Institute for Subatomic Physics, Amsterdam, Netherlands

24Nikhef National Institute for Subatomic Physics and Vrije Universiteit, Amsterdam, Netherlands25Henryk Niewodniczanski Institute of Nuclear Physics Polish Academy of Sciences, Cracow, Poland

26Faculty of Physics & Applied Computer Science, Cracow, Poland27Soltan Institute for Nuclear Studies, Warsaw, Poland

28Horia Hulubei National Institute of Physics and Nuclear Engineering, Bucharest-Magurele, Romania29Petersburg Nuclear Physics Institute (PNPI), Gatchina, Russia

30Institute of Theoretical and Experimental Physics (ITEP), Moscow, Russia31Institute of Nuclear Physics, Moscow State University (SINP MSU), Moscow, Russia

32Institute for Nuclear Research of the Russian Academy of Sciences (INR RAN), Moscow, Russia33Budker Institute of Nuclear Physics (SB RAS) and Novosibirsk State University, Novosibirsk, Russia

34Institute for High Energy Physics (IHEP), Protvino, Russia35Universitat de Barcelona, Barcelona, Spain

36Universidad de Santiago de Compostela, Santiago de Compostela, Spain37European Organization for Nuclear Research (CERN), Geneva, Switzerland38Ecole Polytechnique Federale de Lausanne (EPFL), Lausanne, Switzerland

39Physik-Institut, Universitat Zurich, Zurich, Switzerland40NSC Kharkiv Institute of Physics and Technology (NSC KIPT), Kharkiv, Ukraine

41Institute for Nuclear Research of the National Academy of Sciences (KINR), Kyiv, Ukraine42H.H. Wills Physics Laboratory, University of Bristol, Bristol, United Kingdom43Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom44Department of Physics, University of Warwick, Coventry, United Kingdom

45STFC Rutherford Appleton Laboratory, Didcot, United Kingdom46School of Physics and Astronomy, University of Edinburgh, Edinburgh, United Kingdom

47School of Physics and Astronomy, University of Glasgow, Glasgow, United Kingdom48Oliver Lodge Laboratory, University of Liverpool, Liverpool, United Kingdom

49Imperial College London, London, United Kingdom50School of Physics and Astronomy, University of Manchester, Manchester, United Kingdom

51Department of Physics, University of Oxford, Oxford, United Kingdom52Syracuse University, Syracuse, NY, United States

53CC-IN2P3, CNRS/IN2P3, Lyon-Villeurbanne, France, associated member54Pontifıcia Universidade Catolica do Rio de Janeiro (PUC-Rio), Rio de Janeiro, Brazil, associated to 2

aP.N. Lebedev Physical Institute, Russian Academy of Science (LPI RAS), Moscow, RussiabUniversita di Bari, Bari, Italy

cUniversita di Bologna, Bologna, ItalydUniversita di Cagliari, Cagliari, ItalyeUniversita di Ferrara, Ferrara, ItalyfUniversita di Firenze, Firenze, ItalygUniversita di Urbino, Urbino, Italy

hUniversita di Modena e Reggio Emilia, Modena, ItalyiUniversita di Genova, Genova, Italy

jUniversita di Milano Bicocca, Milano, ItalykUniversita di Roma Tor Vergata, Roma, ItalylUniversita di Roma La Sapienza, Roma, ItalymUniversita della Basilicata, Potenza, Italy

nLIFAELS, La Salle, Universitat Ramon Llull, Barcelona, SpainoInstitucio Catalana de Recerca i Estudis Avancats (ICREA), Barcelona, Spain

pHanoi University of Science, Hanoi, Viet Nam

A model-independent search for direct CP violation in the Cabibbo suppressed decay D+ →K−K+π+ in a sample of approximately 370,000 decays is carried out. The data were collected by theLHCb experiment in 2010 and correspond to an integrated luminosity of 35 pb−1. The normalizedDalitz plot distributions for D+ and D− are compared using four different binning schemes that aresensitive to different manifestations of CP violation. No evidence for CP asymmetry is found.

PACS numbers: 13.25.Ft, 11.30.Er, 14.40.Lb

I. INTRODUCTION

To date CP violation (CPV) has been observed onlyin decays of neutral K and B mesons. All observationsare consistent with CPV being generated by the phase inthe Cabibbo-Kobayashi-Maskawa (CKM) matrix [1, 2] ofthe Standard Model (SM). In the charm sector, CKM dy-namics can produce direct CP asymmetries in Cabibbo

suppressed D± decays of the order of 10−3 or less [3].Asymmetries of up to around 1% can be generated bynew physics (NP) [4, 5]. In most extensions of the SM,asymmetries arise in processes with loop diagrams. How-ever, in some cases CPV could occur even at tree level,for example in models with charged Higgs exchange.

In decays of hadrons, CPV can be observed whentwo different amplitudes with non-zero relative weak

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and strong phases contribute coherently to a final state.Three-body decays are dominated by intermediate reso-nant states, and the requirement of a non-zero relativestrong phase is fulfilled by the phases of the resonances.In two-body decays, CPV leads to an asymmetry in thepartial widths. In three-body decays, the interferencebetween resonances in the two-dimensional phase spacecan lead to observable asymmetries which vary across theDalitz plot.CP -violating phase differences of 10◦ or less do not,

in general, lead to large asymmetries in integrated decayrates, but they could have clear signatures in the Dalitzplot, as we will show in Sect. III. This means that a two-dimensional search should have higher sensitivity than anintegrated measurement. In addition, the distributionof an asymmetry across phase space could hint at theunderlying dynamics.

At present, no theoretical tools for computing decayfractions and relative phases of resonant modes in D de-cays have been applied to multibody D+ decay modes,and no predictions have been made for how asymmetriesmight vary across their Dalitz plots. A full Dalitz plotanalysis of large data samples could, in principle, mea-sure small phase differences. However, rigorous controlof the much larger strong phases would be required. Forthis to be achieved, better understanding of the ampli-tudes, especially in the scalar sector, will be needed, andeffects like three-body final state interactions should betaken into account.

This paper describes a model-independent search fordirect CPV in the Cabibbo suppressed decay D+ →K−K+π+ in a binned Dalitz plot.1 A direct comparisonbetween the D+ and the D− Dalitz plots is made on abin-by-bin basis. The data sample used is approximately35 pb−1 collected in 2010 by the LHCb experiment at acentre of mass energy of

√s = 7 TeV. This data set corre-

sponds to nearly 10 and 20 times more signal events thanused in previous studies of this channel performed by theBABAR [6] and CLEO-c [7] collaborations, respectively.It is comparable to the dataset used in a more recentsearch for CPV in D+ → φπ+ decays at BELLE [8].

The strategy is as follows. For each bin in the Dalitzplot, a local CP asymmetry variable is defined [9, 10],

SiCP =N i(D+)− αN i(D−)√N i(D+) + α2N i(D−)

, α =Ntot(D

+)

Ntot(D−),

(1)where N i(D+) and N i(D−) are the numbers of D± can-didates in the ith bin and α is the ratio between thetotal D+ and D− yields. The parameter α accounts forglobal asymmetries, i.e. those that are constant acrossthe Dalitz plot.

1 Throughout this paper charge conjugation is implied, unless oth-erwise stated.

In the absence of Dalitz plot dependent asymmetries,the SiCP values are distributed according to a Gaussiandistribution with zero mean and unit width. CPV signalsare, therefore, deviations from this behaviour. The nu-merical comparison between the D+ and the D− Dalitzplots is made with a χ2 test (χ2 =

∑(SiCP )2). The num-

ber of degrees of freedom is the number of bins minus one(due to the constraint of the overall D+/D− normaliza-tion). The p-value that results from this test is definedas the probability of obtaining, for a given number of de-grees of freedom and under the assumption of no CPV,a χ2 that is at least as high as the value observed [11].It measures the degree to which we are confident thatthe differences between the D+ and D− Dalitz plots aredriven only by statistical fluctuations.

If CPV is observed, the p-value from this test could beconverted into a significance for a signal using Gaussianstatistics. However, in the event that no CPV is found,there is no model-independent mechanism for setting lim-its on CPV within this procedure. In this case, the resultscan be compared to simulation studies in which an arti-ficial CP asymmetry is introduced into an assumed am-plitude model for the decay. Since such simulations areclearly model-dependent, they are only used as a guideto the sensitivity of the method, and not in the determi-nation of the p-values that constitute the results of theanalysis.

The technique relies on careful accounting for localasymmetries that could be induced by sources such as,the difference in the K–nucleon inelastic cross-section,differences in the reconstruction or trigger efficiencies,left-right detector asymmetries, etc. These effects are in-vestigated in the two control channels D+ → K−π+π+

and D+s → K−K+π+.

The optimum sensitivity is obtained with bins of nearlythe same size as the area over which the asymmetry ex-tends in the Dalitz plot. Since this is a search for newand therefore unknown phenomena, it is necessary to besensitive to effects restricted to small areas as well asthose that extend over a large region of the Dalitz plot.Therefore two types of binning scheme are employed.The first type is simply a uniform grid of equally sizedbins. The second type takes into account the fact that theD+ → K−K+π+ Dalitz plot is dominated by the φπ+

and K∗(892)0K+ resonances, so the event distributionis highly non-uniform. This “adaptive binning” schemeuses smaller bins where the density of events is high, aim-ing for a uniform bin population. In each scheme, differ-ent numbers of bins are used in our search for localizedasymmetries.

The paper is organized as follows. In Sect. II, a descrip-tion of the LHCb experiment and of the data selectionis presented. In Sect. III, the methods and the binningsare discussed in detail. The study of the control channelsand of possible asymmetries generated by detector effectsor backgrounds is presented in Sect. IV. The results ofour search are given in Sect. V, and the conclusions inSect. VI.

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II. DETECTOR, DATASET AND SELECTION

The LHCb detector [12] is a single-arm forward spec-trometer with the main purpose of measuring CPV andrare decays of hadrons containing b and c quarks. A ver-tex locator (VELO) determines with high precision thepositions of the vertices of primary pp collisions (PVs)and the decay vertices of long-lived particles. The track-ing system also includes a large area silicon strip detec-tor located in front of a dipole magnet with an inte-grated field of around 4 Tm, and a combination of siliconstrip detectors and straw drift chambers placed behindthe magnet. Charged hadron identification is achievedwith two ring-imaging Cherenkov (RICH) detectors. Thecalorimeter system consists of a preshower, a scintilla-tor pad detector, an electromagnetic calorimeter and ahadronic calorimeter. It identifies high transverse en-ergy (ET) hadron, electron and photon candidates andprovides information for the trigger. Five muon sta-tions composed of multi-wire proportional chambers andtriple-GEMs (gas electron multipliers) provide fast infor-mation for the trigger and muon identification capability.

The LHCb trigger consists of two levels. The first,hardware-based level selects leptonic and hadronic finalstates with high tranverse momentum, using the subsetof the detectors that are able to reduce the rate at whichthe whole detector is read out to a maximum of 1 MHz.The second level, the High Level Trigger (HLT), is sub-divided into two software stages that can use the infor-mation from all parts of the detector. The first stage,HLT1, performs a partial reconstruction of the event,reducing the rate further and allowing the next stage,HLT2, to fully reconstruct the individual channels. Ateach stage, several selections designed for specific typesof decay exist. As luminosity increased throughout 2010several changes in the trigger were required. To matchthese, the datasets for signal and control modes are di-vided into three parts according to the trigger, samples1, 2 and 3, which correspond to integrated luminositiesof approximately 3, 5 and 28 pb−1 respectively. Themagnet polarity was changed several times during datataking.

The majority of the signal decays come via thehadronic hardware trigger, which has an ET thresholdthat varied between 2.6 and 3.6 GeV in 2010. In theHLT1, most candidates also come from the hadronic se-lections which retain events with at least one high trans-verse momentum (pT) track that is displaced from thePV. In the HLT2, dedicated charm triggers select mostof the signal. However, the signal yield for these channelscan be increased by using other trigger selections, such asthose for decays of the form B → DX. To maintain thenecessary control of Dalitz plot-dependent asymmetries,only events from selections which have been measurednot to introduce charge asymmetries into the Dalitz plotof the D+ → K−π+π+ control mode are accepted.

The signal ( D+ → K−K+π+) and control (D+ →K−π+π+ and D+

s → K−K+π+) mode candidates are

selected using the same criteria, which are chosen to max-imize the statistical significance of the signal. Moreover,care is taken to use selection cuts that do not have a lowefficiency in any part of the Dalitz plot, as this would re-duce the sensitivity in these areas. The selection criteriaare the same regardless of the trigger conditions.

The event selection starts by requiring at least one PVwith a minimum of five charged tracks to exist. To con-trol CPU consumption each event must also have fewerthan 350 reconstructed tracks. The particle identificationsystem constructs a relative log-likelihood for pion andkaon hypotheses, DLLKπ, and we require DLLKπ > 7for kaons and < 3 for pions. Three particles with appro-priate charges are combined to form the D+

(s) candidates.

The corresponding tracks are required to have a goodfit quality (χ2/ndf < 5), pT > 250 MeV/c, momentump > 2000 MeV/c and the scalar sum of their pT above2800 MeV/c. Because a typical D+ travels for around8 mm before decaying, the final state tracks should notpoint to the PV. The smallest displacement from eachtrack to the PV is computed, and a χ2 (χ2

IP), formed byusing the hypothesis that this distance is equal to zero,is required to be greater than 4 for each track. The threedaughters should be produced at a common origin, thecharm decay vertex, with vertex fit χ2/ndf < 10.

This ‘secondary’ vertex must be well separated fromany PV, thus a flight distance variable (χ2

FD) is con-structed. The secondary vertex is required to haveχ2FD > 100, and to be downstream of the PV. The pT

of the D+(s) candidate must be greater than 1000 MeV/c,

and its reconstructed trajectory is required to originatefrom the PV (χ2

IP < 12).In order to quantify the signal yields (S), a simultane-

ous fit to the invariant mass distribution of the D+ andD− samples is performed. A double Gaussian is used forthe K−K+π+ signal, whilst the background (B) is de-scribed by a quadratic component and a single Gaussianfor the small contamination from D∗+ → D0(K−K+)π+

above the D+s peak. The fitted mass spectrum for sam-

ples 1 and 3 combined is shown in Fig. 1, giving theyields shown in Table I. A weighted mean of the widthsof the two Gaussian contributions to the mass peaks isused to determine the overall widths, σ, as 6.35 MeV/c2

for D+ → K−K+π+, 7.05 MeV/c2 for D+s → K−K+π+,

and 8.0 MeV/c2 for D+ → K−π+π+. These values areused to define signal mass windows of approximately 2σin which the Dalitz plots are constructed. The purities,defined as S/(B + S) within these mass regions, are alsoshown in Table I for samples 1 and 3 in the differentdecay modes.

For sample 2, the yield cannot be taken directly fromthe fit, because there is a mass cut in the HLT2 line thataccepts the majority of the signal, selecting events in a±25 MeV/c2 window around the nominal value. How-ever, another HLT2 line with a looser mass cut thatis otherwise identical to the main HLT2 line exists, al-though only one event in 100 is retained. In this line thepurity is found to be the same in sample 2 as in sample

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FIG. 1. Fitted mass spectra of (a) K−π+π+ and (b) K−K+π+ candidates from samples 1 and 3, D+ and D− combined. Thesignal mass windows and sidebands defined in the text are labelled.

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FIG. 2. Dalitz plot of the D+ → K−K+π+ decay for se-lected candidates in the signal window. The vertical K∗(892)0

and horizontal φ(1020) contributions are clearly visible in thedata.

3. The yield in sample 2 is then inferred as the total(S +B) in all allowed triggers in the mass window timesthe purity in sample 3. Thus the overall yield of signalD+ → K−K+π+ candidates in the three samples withinthe mass window is approximately 370,000. The totalnumber of candidates (S + B) in each decay mode usedin the analysis are given in Table II. The Dalitz plot ofdata in the D+ window is shown in Fig. 2.

Within the 2σ D+ → K−K+π+ mass window, about8.6% of events are background. Apart from randomthree-body track combinations, charm backgrounds andtwo-body resonances plus one track are expected. Charmreflections appear when a particle is wrongly identified ina true charm three-body decay and/or a track in a four-body charm decay is lost. The main three-body reflec-tion in the K−K+π+ spectrum is the Cabibbo-favoured

TABLE I. Yield (S) and purity for samples 1 and 3 afterthe final selection. The purity is estimated in the 2σ masswindow.

Decay Yield PuritySample 1+3 Sample 1 Sample 3

D+ → K−K+π+ (3.284± 0.006)× 105 88% 92%D+

s → K−K+π+ (4.615± 0.012)× 105 89% 92%D+ → K−π+π+ (3.3777± 0.0037)× 106 98% 98%

TABLE II. Number of candidates (S + B) in the signal win-dows shown in Fig. 1 after the final selection, for use in thesubsequent analysis.

sample 1 sample 2 sample 3 TotalD+ → K−K+π+ 84,667 65,781 253,446 403,894D+

s → K−K+π+ 126,206 91,664 346,068 563,938D+ → K−π+π+ 858,356 687,197 2,294,315 3,839,868

D+ → K−π+π+, where the incorrect assignment of thekaon mass to the pion leads to a distribution that par-tially overlaps with the D+

s → K−K+π+ signal region,but not with D+ → K−K+π+. The four body, Cabibbo-favoured mode D0 → K−π+π−π+ where a π+ is lostand the π− is misidentified as a K− will appear broadlydistributed in K−K+π+ mass, but its resonances couldcreate structures in the Dalitz plot. Similarly, K∗(892)0

and φ resonances from the PV misreconstructed with arandom track forming a three-body vertex will also ap-pear.

7

TABLE III. The CLEO-c amplitude model “B” [7] used in the simulation studies. The uncertainties are statistical, experimentalsystematic and model systematic respectively.

Resonance Amplitude Relative phase Fit fraction

K∗(892)0 1 (fixed) 0 (fixed) 25.7± 0.5+0.4+0.1−0.3−1.2

K∗0(1430)0 4.56± 0.13+0.10+0.42

−0.01−0.39 70± 6+1+16−6−23 18.8± 1.2+0.6+3.2

−0.1−3.4

κ(800) 2.30± 0.13+0.01+0.52−0.11−0.29 − 87± 6+2+15

−3−10 7.0± 0.8+0.0+3.5−0.6−1.9

K∗2(1430)0 7.6± 0.8+0.5+2.4

−0.6−4.8 171± 4+0+24−2−11 1.7± 0.4+0.3+1.2

−0.2−0.7

φ(1020) 1.166± 0.015+0.001+0.025−0.009−0.009 −163± 3+1+14

−1−5 27.8± 0.4+0.1+0.2−0.3−0.4

a0(1450)0 1.50± 0.10+0.09+0.92−0.06−0.33 116± 2+1+7

−1−14 4.6± 0.6+0.5+7.2−0.3−1.8

φ(1680) 1.86± 0.20+0.02+0.62−0.08−0.77 −112± 6+3+19

−4−12 0.51± 0.11+0.01+0.37−0.04−0.15

III. METHODS AND BINNINGS

Monte Carlo pseudo-experiments are used to verifythat we can detect CPV with the strategy outlined inSect. I without producing fake signals, and to devise andtest suitable binning schemes for the Dalitz plot. Theyare also used to quantify our sensitivity to possible man-ifestations of CPV, where we define the sensitivity to agiven level of CPV as the probability of observing it with3σ significance.

For the D+ → K−K+π+ Dalitz plot model, the resultof the CLEO-c analysis (fit B) [7] is used. The ampli-tudes and phases of the resonances used in this modelare reproduced in Table III. For simplicity, only resonantmodes with fit fractions greater than 2% are includedin the pseudo-experiments. The fit fraction for a reso-nance is defined as the integral of its squared amplitudeover the Dalitz plot divided by the integral of the squareof the overall complex amplitude. An efficiency func-tion is determined from a two-dimensional second orderpolynomial fit to the Dalitz plot distribution of triggeredevents that survive the selection cuts in the GEANT-based [13] LHCb Monte Carlo simulation for nonresonantD+ → K−K+π+. A simple model for the backgroundis inferred from the Dalitz plots of the sidebands of theD+ → K−K+π+ signal. It is composed of random com-binations of K−, K+, and π+ tracks, φ resonances withπ+ tracks, and K∗(892)0 resonances with K+ tracks.The CLEO-c Dalitz plot analysis has large uncertainties,as do the background and efficiency simulations (due tolimited numbers of MC events), so the method is testedon a range of different Dalitz plot models.

Pseudo-experiments with large numbers of events areused to investigate how CPV would be observed in theDalitz plot. These experiments are simple “toy” simu-lations that produce points in the Dalitz plot accordingto the probability density function determined from theCLEO-c amplitude model with no representation of theproton-proton collision, detector, or trigger. Figure 3(a)illustrates the values of SiCP observed with 8×107 eventsand no CPV. This dataset is approximately 50 timeslarger than the data sample under study. The result-ing χ2/ndf is 253.4/218, giving a p-value for consistencywith no CPV of 5.0%. This test shows that the method

by itself is very unlikely to yield false positive results.Figure 3(b) shows an example test using 5 × 107 eventswith a CP violating phase difference of 4◦ between theamplitudes for the φ(1020)π+ component in D+ and D−

decays. The p-value in this case is less than 10−100. TheCPV effect is clearly visible, and is spread over a broadarea of the plot, changing sign from left to right. Thissign change means the CPV causes only a 0.1% differ-ence in the total decay rate between D+ and D−. Thisillustrates the strength of our method, as the asymme-try would be much more difficult to detect in a measure-ment that was integrated over the Dalitz plot. Even withno systematic uncertainties, to see a 0.1% asymmetry atthe 3σ level would require 2.25 × 106 events. With themethod and much smaller dataset used here we wouldobserve this signal at the 3σ level with 76% probability,as shown in Table IV below.

The sensitivity to a particular manifestation of CPVdepends on the choice of binning. The fact that the CP -violating region in most of the pseudo-experiments coversa broad area of the Dalitz plot suggests that the optimalnumber of bins for this type of asymmetry is low. Eachbin adds a degree of freedom without changing the χ2

value for consistency with no CPV. However, if CP asym-metries change sign within a bin, they will not be seen.Similarly, the sensitivity is reduced if only a small partof a large bin has any CPV in it. To avoid effects dueto excessive fluctuations, bins that contain fewer than 50candidates are not used anywhere in the analysis. Suchbins are very rare.

The binnings are chosen to reflect the highly non-uniform structure of the Dalitz plot. A simple adap-tive binning algorithm was devised to define binnings ofapproximately equal population without separating D+

and D−. Two binnings that are found to have goodsensitivity to the simulated asymmetries contain 25 bins(“Adaptive I”) arranged as shown in Fig. 4(a), and 106bins (“Adaptive II”) arranged as shown in Fig. 4(b). ForAdaptive I, a simulation of the relative value of the strongphase across the Dalitz plot in the CLEO-c amplitudemodel is used to refine the results of the algorithm: ifthe strong phase varies significantly across a bin, CPasymmetries are more likely to change sign. Thereforethe bin boundaries are adjusted to minimise changes in

8

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FIG. 3. SCP across the Dalitz plot in a Monte Carlo pseudo-experiment with a large number of events with (a) no CPV and(b) a 4◦ CPV in the φπ phase. Note the difference in colour scale between (a) and (b).

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FIG. 4. Layout of the (a) “Adaptive I” and (b) “Adaptive II” binnings on the Dalitz plot of data.

the strong phase within bins. The model-dependence ofthis simulation could, in principle, influence the binningand therefore the sensitivity to CPV, but it cannot intro-duce model-dependence into the final results as no artifi-cial signal could result purely from the choice of binning.Two further binning schemes, “Uniform I” and “UniformII”, are defined. These use regular arrays of rectangularbins of equal size.

The adaptive binnings are used to determine the sen-sitivity to several manifestations of CPV. With 200 testexperiments of approximately the same size as the sig-nal sample in data, including no asymmetries, no CP -violating signals are observed at the 3σ level with Adap-tive I or Adaptive II. The expectation is 0.3.

With the chosen binnings, a number of sets of 100pseudo-experiments with different CP -violating asymme-tries are produced. The probability of observing a givensignal in either the φ(1020) or κ(800) resonances with3σ significance is calculated in samples of the same sizeas the dataset. The results are given in Table IV. TheCPV shows up both in the χ2/ndf and in the width of

the fitted SCP distribution.For comparison, the asymmetries in the φ phase and

κ magnitude measured by the CLEO collaboration us-ing the same amplitude model were (6 ± 6+0+6

−2−2)◦ and

(−12± 12+6+2−1−10)%,2 where the uncertainties are statisti-

cal, systematic and model-dependent, respectively. Ta-ble IV suggests that, assuming their model, we would beat least 95% confident of detecting the central values ofthese asymmetries.

The sensitivity of the results to variations in the Dalitzplot model and the background is investigated, and ex-ample results for the CP asymmetry in the φ(1020) phaseare shown in Table V. In this table, models A and Bare taken from the CLEO paper, model B2 includes anf0(980) contribution that accounts for approximately 8%

2 The conventions used in the CLEO paper to define asymmetryare different, so the asymmetries in Table II of [7] have beenmultiplied by two in order to be comparable with those givenabove.

9

TABLE IV. Results from sets of 100 pseudo-experiments with different CP asymmetries and Adaptive I and II binnings. p(3σ)is the probability of a 3σ observation of CPV. 〈S〉 is the mean significance with which CPV is observed.

CPV Adaptive I Adaptive IIp(3σ) 〈S〉 p(3σ) 〈S〉

no CPV 0 0.84σ 1% 0.84σ6◦ in φ(1020) phase 99% 7.0σ 98% 5.2σ5◦ in φ(1020) phase 97% 5.5σ 79% 3.8σ4◦ in φ(1020) phase 76% 3.8σ 41% 2.7σ3◦ in φ(1020) phase 38% 2.8σ 12% 1.9σ2◦ in φ(1020) phase 5% 1.6σ 2% 1.2σ

6.3% in κ(800) magnitude 16% 1.9σ 24% 2.2σ11% in κ(800) magnitude 83% 4.2σ 95% 5.6σ

TABLE V. Results from sets of 100 pseudo-experiments with 4◦ CPV in the φ(1020) phase and different Dalitz plot models.p(3σ) is the probability of a 3σ observation of CPV. 〈S〉 is the mean significance with which CPV is observed. The sample sizeis comparable to that seen in data.

Model Adaptive I Adaptive IIp(3σ) 〈S〉 p(3σ) 〈S〉

B (baseline) 76% 3.8σ 41% 2.7σA 84% 4.3σ 47% 2.9σ

B2 (add f0(980)) 53% 3.2σ 24% 2.2σB3 (vary K∗

0(1430)0 magn.) 82% 4.0σ 41% 2.8σB4 (vary K∗

0(1430)0 phase) 73% 3.7σ 38% 2.7σ

of events, and models B3 and B4 are variations of theK∗

0(1430)0 amplitude and phase within their uncertain-ties. As expected, the sensitivity to CPV in the reso-nances of an amplitude model depends quite strongly onthe details of the model. This provides further justifi-cation for our model-independent approach. However, areasonable level of sensitivity is retained in all the caseswe tested. Thus, when taken together, the studies showthat the method works well. It does not yield fake signals,and should be sensitive to any large CPV that varies sig-nificantly across the Dalitz plot even if it does not occurprecisely in the way investigated here.

IV. CONTROL MODES

It is possible that asymmetries exist in the data thatdo not result from CPV, for example due to production,backgrounds, instrumental effects such as left-right dif-ferences in detection efficiency, or momentum-dependentdifferences in the interaction cross-sections of the daugh-ter particles with detector material. Our sensitivity tosuch asymmetries is investigated in the two Cabibbofavoured control channels, where there is no large CPVpredicted. The D+ → K−π+π+ control mode has anorder of magnitude more candidates than the Cabibbo-suppressed signal mode, and is more sensitive to detec-tor effects since there is no cancellation between K+ andK− reconstruction efficiencies. Conversely, the D+

s →

K−K+π+ control mode is very similar to our signal modein terms of resonant structure, number of candidates,kinematics, detector effects, and backgrounds.

The control modes and their mass sidebands definedin Fig. 1 are tested for asymmetries using the methoddescribed in the previous section. Adaptive and uniformbinning schemes are defined for D+ → K−π+π+ andD+s → K−K+π+. They are applied to samples 1–3 and

each magnet polarity separately. In the final results, theasymmetries measured in data taken with positive andnegative magnet polarity are combined in order to cancelleft-right detector asymmetries. The precise number ofbins chosen is arbitrary, but care is taken to use a widerange of tests with binnings that reflect the size of thedataset for the decay mode under study.

For D+ → K−π+π+, five different sets of bins in eachscheme are used. A very low p-value would indicate alocal asymmetry. One test with 25 adaptive bins in oneof the subsamples (with negative magnet polarity) hasa p-value of 0.1%, but when combined with the positivepolarity sample the p-value increases to 1.7%. All othertests yield p-values ranging from 1–98%. Some exampleresults are given in Table VI. A typical distribution ofthe SCP values with a Gaussian fit is shown in Fig. 5(a)for a test with 900 uniform bins. The fitted values ofthe mean and width are consistent with one and zerorespectively, suggesting that the differences between theD+ and the D− Dalitz plots are driven only by statisticalfluctuations.

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FIG. 5. (a) Distribution of SCP values from D+ → K−π+π+ from a test with 900 uniform bins. The mean of the fittedGaussian distribution is 0.015 ± 0.034 and the width is 0.996 ± 0.023. (b) Distribution of SCP values from D+

s → K−K+π+

with 129 bins. The fitted mean is −0.011± 0.084 and the width is 0.958± 0.060.

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FIG. 6. Dalitz plots of (a) D+ → K−π+π+, showing the 25-bin adaptive scheme with the SCP values, and (b) D+s → K−K+π+,

showing the three regions referred to in the text. The higher and lower K−π+ invariant mass combinations are plotted in (a)as there are identical pions in the final state.

TABLE VI. Results (p-values, in %) from tests with theD+ → K−π+π+ control channel using the uniform and adap-tive binning schemes. The values correspond to tests per-formed on the whole dataset in the mass windows defined inSect. II.

1300 bins 900 bins 400 bins 100 bins 25 binsUniform 73.8 17.7 72.6 54.6 1.7Adaptive 81.7 57.4 65.8 30.0 11.8

For the D+s → K−K+π+ mode a different procedure

is followed due to the smaller sample size and to the highdensity of events along the φ and the K∗(892)0 bands.The Dalitz plot is divided into three zones, as shown inFig. 6. Each zone is further divided into 300, 100 and 30bins of same size. The results are given in Table VII. Inaddition, a test is performed on the whole Dalitz plot us-

TABLE VII. Results (p-values, in %) from tests with theD+

s → K−K+π+ control channel using the uniform binningscheme. The values correspond to tests performed separatelyon Zones A-C, with samples 1-3 and both magnet polaritiescombined.

bins Zone A Zone B Zone C300 20.1 25.3 14.5100 41.7 84.6 89.530 66.0 62.5 24.6

ing 129 bins chosen by the adaptive algorithm, and a ver-sion of the 25-bin scheme outlined in Sect. III scaled bythe ratio of the available phase space in the two modes.These tests yield p-values of 71.5% and 34.3% respec-tively.

Other possible sources of local charge asymmetry in the

11

TABLE VIII. Results (p-values, in %) from tests with theD+ → K−π+π+ and D+

s → K−K+π+ samples divided ac-cording to the impact parameter with respect to the primaryvertex. The tests are performed using the adaptive binningscheme with 25 bins.

χ2IP < 6 χ2

IP > 6D+ → K−π+π+ 8.5 88.9D+

s → K−K+π+ 52.0 30.6

signal region are the charm contamination of the back-ground, and asymmetries from CPV in misreconstructedB decays. In order to investigate the first possibility,similar tests are carried out in the mass sidebands of theD+

(s) → K−K+π+ signal (illustrated in Fig. 1). There is

no evidence for asymmetries in the background.From a simulation of the decay D+ → K−π+π+ the

level of secondary charm (B → DX) in our selected sam-ple is found to be 4.5%. The main discriminating vari-able to distinguish between prompt and secondary charmis the impact parameter (IP) of the D with respect to theprimary vertex. Given the long B lifetime, the IP distri-bution of secondary charm candidates is shifted towardslarger values compared to that of prompt D+ mesons.

The effect of secondary charm is investigated by divid-ing the data set according to the value of the candidateIP significance (χ2

IP ). The subsample with events havinglarger χ2

IP are likely to be richer in secondary charm. Theresults are shown in Table VIII. No anomalous effects areseen in the high χ2

IP sample, so contamination from sec-ondary charm with CPV does not affect our results forstudies with our current level of sensitivity.

The analysis on the two control modes and on thesidebands in the final states K−K+π+ and K−π+π+

gives results from all tests that are fully consistent withno asymmetry. Therefore, any asymmetry observed inD+ → K−K+π+ is likely to be a real physics effect.

V. RESULTS

The signal sample with which we search for CP vio-lation consists of 403,894 candidates selected within theK−K+π+ mass window from 1856.7 to 1882.1 MeV/c2,as described in Sect. II. There are 200,336 and 203,558D+ and D− candidates respectively. This implies a nor-malization factor α = Ntot(D

+)/Ntot(D−) = 0.984 ±

0.003, to be used in Eq. 1.The strategy for looking for signs of localized CPV is

discussed in the previous sections. In the absence of localasymmetries in the control channels D+ → K−π+π+ andD+s → K−K+π+ and in the sidebands of the K−K+π+

mass spectrum, we investigate the signal sample underdifferent binning choices.

First, the adaptive binning is used with 25 and 106bins in the Dalitz plot as illustrated in Fig. 4. Then

CPV is investigated with uniform binnings, using 200and 530 bins of equal size. For each of these binningchoices, the significance SiCP of the difference in D+ andD− population is computed for each bin i, as defined inEq. 1. The χ2/ndf =

∑i(SiCP )2/ndf is calculated and

the p-value is obtained. The distributions of SiCP arefitted to Gaussian functions.

The p-values are shown in Table IX. The Dalitz plotdistributions of SiCP are shown in Fig. 7. In Fig. 8 the dis-tributions of SiCP and the corresponding Gaussian fits forthe different binnings are shown. The p-values obtainedindicate no evidence for CPV. This is corroborated bythe good fits of the SiCP distributions to Gaussians, withmeans and widths consistent with 0 and 1, respectively.

As further checks, many other binnings are tested.The number of bins in the adaptive and uniform bin-ning schemes is varied from 28 to 106 and from 21 to530 respectively. The samples are separated accordingto the magnet polarity and the same studies are re-peated. In all cases the p-values are consistent with noCPV, with values ranging from 4% to 99%. We concludethat there is no evidence for CPV in our data sample ofD+ → K−K+π+.

VI. CONCLUSION

Due to the rich structure of their Dalitz plots, threebody charm decays are sensitive to CP violating phaseswithin and beyond the Standard Model. Here, a model-independent search for direct CP violation is performedin the Cabibbo suppressed decay D+ → K−K+π+ with35 pb−1 of data collected by the LHCb experiment, andno evidence for CPV is found. Several binnings are usedto compare normalised D+ and D− Dalitz plot distri-butions. This technique is validated with large num-bers of simulated pseudo-experiments and with Cabibbofavoured control channels from the data: no false positivesignals are seen. To our knowledge this is the first timea search for CPV is performed using adaptive bins whichreflect the structure of the Dalitz plot.

Monte Carlo simulations illustrate that large localisedasymmetries can occur without causing detectable differ-ences in integrated decay rates. The technique used hereis shown to be sensitive to such asymmetries. Assumingthe decay model, efficiency parameterisation and back-ground model described in Sect. III we would be 90%confident of seeing a CP violating difference of either 5◦

in the phase of the φπ+ or 11% in the magnitude of theκ(800)K+ with 3σ significance. Since we find no evidenceof CPV, effects of this size are unlikely to exist.

VII. ACKNOWLEDGMENTS

We express our gratitude to our colleagues in theCERN accelerator departments for the excellent per-formance of the LHC. We thank the technical and ad-

12

TABLE IX. Fitted means and widths, χ2/ndf and p-values for consistency with no CPV for the D+ → K−K+π+ decay modewith four different binnings.

Binning Fitted mean Fitted width χ2/ndf p-value (%)Adaptive I 0.01± 0.23 1.13± 0.16 32.0/24 12.7Adaptive II −0.024± 0.010 1.078± 0.074 123.4/105 10.6Uniform I −0.043± 0.073 0.929± 0.051 191.3/198 82.1Uniform II −0.039± 0.045 1.011± 0.034 519.5/529 60.5

)4/c2 (GeV2+π-Km

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FIG. 7. Distribution of SiCP in the Dalitz plot for (a) “Adaptive I”, (b) “Adaptive II”, (c) “Uniform I” and (d) “Uniform II”.

In (c) and (d) bins at the edges are not shown if the number of entries is not above a threshold of 50 (see Sect. III).

ministrative staff at CERN and at the LHCb insti-tutes, and acknowledge support from the National Agen-cies: CAPES, CNPq, FAPERJ and FINEP (Brazil);CERN; NSFC (China); CNRS/IN2P3 (France); BMBF,DFG, HGF and MPG (Germany); SFI (Ireland); INFN(Italy); FOM and NWO (Netherlands); SCSR (Poland);

ANCS (Romania); MinES of Russia and Rosatom (Rus-sia); MICINN, XuntaGal and GENCAT (Spain); SNSFand SER (Switzerland); NAS Ukraine (Ukraine); STFC(United Kingdom); NSF (USA). We also acknowledgethe support received from the ERC under FP7 and theRegion Auvergne.

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