LHCb-PAPER-2011-017 CERN-PH-EP-2011-163 November 2, 2018
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. Artamonov34, 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. B¨uchler-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 Su´arez36,
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. Est`eve43, A. Falabella16,e, E. Fanchini20,j, C. F¨arber11, 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. G¨obel54, D. Golubkov30, A. Golutvin49,30,37, A. Gomes2, H. Gordon51,
M. Grabalosa G´andara35, R. Graciani Diaz35, L.A. Granado Cardoso37, E. Graug´es35, 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. Lef`evre5, A. Leflat31,37, J. Lefran¸cois7, 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. M¨arki38, J. Marks11, G. Martellotti22,
A. Martens7, L. Martin51, A. Mart´ın S´anchez7, D. Martinez Santos37, D. Martins Tostes1, A. Massafferri1,
Z. Mathe12, C. Matteuzzi20, M. Matveev29, E. Maurice6, B. Maynard52, A. Mazurov16,32,37, G. McGregor50, 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. M¨uller39, 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. P´erez-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. Zvyagin37.
1Centro Brasileiro de Pesquisas F´ısicas (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, Universit´e de Savoie, CNRS/IN2P3, Annecy-Le-Vieux, France
5Clermont Universit´e, Universit´e Blaise Pascal, CNRS/IN2P3, LPC, Clermont-Ferrand, France 6CPPM, Aix-Marseille Universit´e, CNRS/IN2P3, Marseille, France
7LAL, Universit´e Paris-Sud, CNRS/IN2P3, Orsay, France
8LPNHE, Universit´e Pierre et Marie Curie, Universit´e Paris Diderot, CNRS/IN2P3, Paris, France 9Fakult¨at Physik, Technische Universit¨at Dortmund, Dortmund, Germany
10Max-Planck-Institut f¨ur Kernphysik (MPIK), Heidelberg, Germany 11Physikalisches Institut, Ruprecht-Karls-Universit¨at Heidelberg, Heidelberg, Germany
12School of Physics, University College Dublin, Dublin, Ireland 13Sezione INFN di Bari, Bari, Italy
14Sezione INFN di Bologna, Bologna, Italy 15Sezione INFN di Cagliari, Cagliari, Italy 16Sezione INFN di Ferrara, Ferrara, Italy 17Sezione INFN di Firenze, Firenze, Italy 18Laboratori Nazionali dell’INFN di Frascati, Frascati, Italy
19Sezione INFN di Genova, Genova, Italy 20Sezione INFN di Milano Bicocca, Milano, Italy 21Sezione INFN di Roma Tor Vergata, Roma, Italy
22Sezione INFN di Roma La Sapienza, Roma, Italy
23Nikhef National Institute for Subatomic Physics, Amsterdam, Netherlands
24Nikhef National Institute for Subatomic Physics and Vrije Universiteit, Amsterdam, Netherlands 25Henryk Niewodniczanski Institute of Nuclear Physics Polish Academy of Sciences, Cracow, Poland
26Faculty of Physics & Applied Computer Science, Cracow, Poland 27Soltan Institute for Nuclear Studies, Warsaw, Poland
28Horia Hulubei National Institute of Physics and Nuclear Engineering, Bucharest-Magurele, Romania 29Petersburg Nuclear Physics Institute (PNPI), Gatchina, Russia
30Institute of Theoretical and Experimental Physics (ITEP), Moscow, Russia 31Institute of Nuclear Physics, Moscow State University (SINP MSU), Moscow, Russia 32Institute for Nuclear Research of the Russian Academy of Sciences (INR RAN), Moscow, Russia 33Budker Institute of Nuclear Physics (SB RAS) and Novosibirsk State University, Novosibirsk, Russia
34Institute for High Energy Physics (IHEP), Protvino, Russia 35Universitat de Barcelona, Barcelona, Spain
36Universidad de Santiago de Compostela, Santiago de Compostela, Spain 37European Organization for Nuclear Research (CERN), Geneva, Switzerland
38Ecole Polytechnique F´ed´erale de Lausanne (EPFL), Lausanne, Switzerland 39Physik-Institut, Universit¨at Z¨urich, Z¨urich, Switzerland
40NSC Kharkiv Institute of Physics and Technology (NSC KIPT), Kharkiv, Ukraine 41Institute for Nuclear Research of the National Academy of Sciences (KINR), Kyiv, Ukraine
42H.H. Wills Physics Laboratory, University of Bristol, Bristol, United Kingdom 43Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom 44Department of Physics, University of Warwick, Coventry, United Kingdom
45STFC Rutherford Appleton Laboratory, Didcot, United Kingdom
46School of Physics and Astronomy, University of Edinburgh, Edinburgh, United Kingdom 47School of Physics and Astronomy, University of Glasgow, Glasgow, United Kingdom
48Oliver Lodge Laboratory, University of Liverpool, Liverpool, United Kingdom 49Imperial College London, London, United Kingdom
50School of Physics and Astronomy, University of Manchester, Manchester, United Kingdom 51Department of Physics, University of Oxford, Oxford, United Kingdom
52Syracuse University, Syracuse, NY, United States
53CC-IN2P3, CNRS/IN2P3, Lyon-Villeurbanne, France, associated member
54Pontif´ıcia Universidade Cat´olica do Rio de Janeiro (PUC-Rio), Rio de Janeiro, Brazil, associated to2
aP.N. Lebedev Physical Institute, Russian Academy of Science (LPI RAS), Moscow, Russia bUniversit`a di Bari, Bari, Italy
cUniversit`a di Bologna, Bologna, Italy dUniversit`a di Cagliari, Cagliari, Italy eUniversit`a di Ferrara, Ferrara, Italy fUniversit`a di Firenze, Firenze, Italy gUniversit`a di Urbino, Urbino, Italy
hUniversit`a di Modena e Reggio Emilia, Modena, Italy iUniversit`a di Genova, Genova, Italy jUniversit`a di Milano Bicocca, Milano, Italy kUniversit`a di Roma Tor Vergata, Roma, Italy lUniversit`a di Roma La Sapienza, Roma, Italy mUniversit`a della Basilicata, Potenza, Italy
nLIFAELS, La Salle, Universitat Ramon Llull, Barcelona, Spain oInstituci´o Catalana de Recerca i Estudis Avan¸cats (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 the
LHCb experiment in 2010 and correspond to an integrated luminosity of 35 pb−1. The normalized
Dalitz plot distributions for D+and D−
are compared using four different binning schemes that are sensitive 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 only in decays of neutral K and B mesons. All observations are consistent with CPV being generated by the phase in the Cabibbo-Kobayashi-Maskawa (CKM) matrix [1, 2] of the 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 by new 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 when two different amplitudes with non-zero relative weak
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 relative strong phase is fulfilled by the phases of the resonances. In two-body decays, CPV leads to an asymmetry in the partial widths. In three-body decays, the interference between resonances in the two-dimensional phase space can lead to observable asymmetries which vary across the Dalitz plot.
CP -violating phase differences of 10◦ or less do not, in general, lead to large asymmetries in integrated decay rates, but they could have clear signatures in the Dalitz plot, as we will show in Sect. III. This means that a two-dimensional search should have higher sensitivity than an integrated measurement. In addition, the distribution of an asymmetry across phase space could hint at the underlying dynamics.
At present, no theoretical tools for computing decay fractions 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 asymmetries might vary across their Dalitz plots. A full Dalitz plot analysis of large data samples could, in principle, mea-sure small phase differences. However, rigorous control of the much larger strong phases would be required. For this to be achieved, better understanding of the ampli-tudes, especially in the scalar sector, will be needed, and effects like three-body final state interactions should be taken into account.
This paper describes a model-independent search for direct CPV in the Cabibbo suppressed decay D+ → K−K+π+in a binned Dalitz plot.1 A direct comparison
between the D+ and the D− Dalitz plots is made on a
bin-by-bin basis. The data sample used is approximately 35 pb−1 collected in 2010 by the LHCb experiment at a
centre of mass energy of√s = 7 TeV. This data set corre-sponds to nearly 10 and 20 times more signal events than used in previous studies of this channel performed by the BABAR [6] and CLEO-c [7] collaborations, respectively. It is comparable to the dataset used in a more recent search for CPV in D+→ φπ+ decays at BELLE [8].
The strategy is as follows. For each bin in the Dalitz plot, a local CP asymmetry variable is defined [9, 10],
SCPi = Ni(D+) − αNi(D−) pNi(D+) + α2Ni(D−) , α = Ntot(D+) Ntot(D−) , (1) where Ni(D+) and Ni(D−) are the numbers of D±
can-didates in the ith bin and α is the ratio between the
total D+ and D− yields. The parameter α accounts for global asymmetries, i.e. those that are constant across the Dalitz plot.
1 Throughout this paper charge conjugation is implied, unless oth-erwise stated.
In the absence of Dalitz plot dependent asymmetries, the Si
CP values are distributed according to a Gaussian
distribution with zero mean and unit width. CPV signals are, therefore, deviations from this behaviour. The nu-merical comparison between the D+ and the D− Dalitz
plots is made with a χ2test (χ2=P(Si CP)
2). The
num-ber of degrees of freedom is the numnum-ber of bins minus one (due to the constraint of the overall D+/D−
normaliza-tion). The p-value that results from this test is defined as 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 that the differences between the D+ and D− Dalitz plots are
driven only by statistical fluctuations.
If CPV is observed, the p-value from this test could be converted into a significance for a signal using Gaussian statistics. 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 results can 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 are clearly model-dependent, they are only used as a guide to the sensitivity of the method, and not in the determi-nation of the p-values that constitute the results of the analysis.
The technique relies on careful accounting for local asymmetries 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 nearly the same size as the area over which the asymmetry ex-tends in the Dalitz plot. Since this is a search for new and therefore unknown phenomena, it is necessary to be sensitive to effects restricted to small areas as well as those 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 sized bins. The second type takes into account the fact that the D+ → K−K+π+ Dalitz plot is dominated by the φπ+
and K∗(892)0K+ resonances, so the event distribution
is highly non-uniform. This “adaptive binning” scheme uses 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 localized asymmetries.
The paper is organized as follows. In Sect. II, a descrip-tion of the LHCb experiment and of the data selecdescrip-tion is presented. In Sect. III, the methods and the binnings are discussed in detail. The study of the control channels and of possible asymmetries generated by detector effects or backgrounds is presented in Sect. IV. The results of our search are given in Sect. V, and the conclusions in Sect. VI.
II. DETECTOR, DATASET AND SELECTION The LHCb detector [12] is a single-arm forward spec-trometer with the main purpose of measuring CPV and rare decays of hadrons containing b and c quarks. A ver-tex locator (VELO) determines with high precision the positions 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 silicon strip detectors and straw drift chambers placed behind the magnet. Charged hadron identification is achieved with two ring-imaging Cherenkov (RICH) detectors. The calorimeter system consists of a preshower, a scintilla-tor pad detecscintilla-tor, an electromagnetic calorimeter and a hadronic calorimeter. It identifies high transverse en-ergy (ET) hadron, electron and photon candidates and
provides information for the trigger. Five muon sta-tions composed of multi-wire proportional chambers and triple-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 final states with high tranverse momentum, using the subset of the detectors that are able to reduce the rate at which the 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. At each stage, several selections designed for specific types of decay exist. As luminosity increased throughout 2010 several changes in the trigger were required. To match these, the datasets for signal and control modes are di-vided into three parts according to the trigger, samples 1, 2 and 3, which correspond to integrated luminosities of approximately 3, 5 and 28 pb−1 respectively. The magnet polarity was changed several times during data taking.
The majority of the signal decays come via the hadronic hardware trigger, which has an ET threshold
that varied between 2.6 and 3.6 GeV in 2010. In the HLT1, 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 the
PV. In the HLT2, dedicated charm triggers select most of the signal. However, the signal yield for these channels can be increased by using other trigger selections, such as those for decays of the form B → DX. To maintain the necessary control of Dalitz plot-dependent asymmetries, only events from selections which have been measured not to introduce charge asymmetries into the Dalitz plot of 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 low efficiency in any part of the Dalitz plot, as this would re-duce the sensitivity in these areas. The selection criteria are the same regardless of the trigger conditions.
The event selection starts by requiring at least one PV with a minimum of five charged tracks to exist. To con-trol CPU consumption each event must also have fewer than 350 reconstructed tracks. The particle identification system constructs a relative log-likelihood for pion and kaon hypotheses, DLLKπ, and we require DLLKπ > 7
for 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 good fit quality (χ2/ndf < 5), p
T > 250 MeV/c, momentum
p > 2000 MeV/c and the scalar sum of their pT above
2800 MeV/c. Because a typical D+ travels for around
8 mm before decaying, the final state tracks should not point to the PV. The smallest displacement from each track to the PV is computed, and a χ2 (χ2
IP), formed by
using the hypothesis that this distance is equal to zero, is required to be greater than 4 for each track. The three daughters should be produced at a common origin, the charm decay vertex, with vertex fit χ2/ndf < 10.
This ‘secondary’ vertex must be well separated from any PV, thus a flight distance variable (χ2
FD) is
con-structed. The secondary vertex is required to have χ2FD > 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 originate from the PV (χ2IP< 12).
In order to quantify the signal yields (S), a simultane-ous fit to the invariant mass distribution of the D+ and
D− samples is performed. A double Gaussian is used for
the K−K+π+ signal, whilst the background (B) is de-scribed by a quadratic component and a single Gaussian for 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 the yields shown in Table I. A weighted mean of the widths of the two Gaussian contributions to the mass peaks is used to determine the overall widths, σ, as 6.35 MeV/c2
for D+→ K−K+π+, 7.05 MeV/c2for D+
s → K−K+π+,
and 8.0 MeV/c2 for D+ → K−π+π+. These values are
used to define signal mass windows of approximately 2σ in which the Dalitz plots are constructed. The purities, defined as S/(B + S) within these mass regions, are also shown in Table I for samples 1 and 3 in the different decay modes.
For sample 2, the yield cannot be taken directly from the fit, because there is a mass cut in the HLT2 line that accepts 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 that is otherwise identical to the main HLT2 line exists, al-though only one event in 100 is retained. In this line the purity is found to be the same in sample 2 as in sample
) 2 (MeV/c + π + π -K m 1800 1850 1900 ) 2 Events / ( 0.28 MeV/c 0 20000 40000 lower upper +
D
LHCb
(a)
) 2 (MeV/c + π + K -K m 1800 1850 1900 1950 2000 ) 2 Events / ( 0.48 MeV/c 0 5000 10000 15000lower middle upper
+
D
D
+sLHCb
(b)
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. The signal mass windows and sidebands defined in the text are labelled.
) 4 /c 2 (GeV 2 + π -K m 0.5 1 1.5 2 ) 4 /c 2 (GeV 2 + K -K m 1 1.5 2 2.5 3 1 10 2 10 3 10
LHCb
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 the data.
3. The yield in sample 2 is then inferred as the total (S + B) in all allowed triggers in the mass window times the purity in sample 3. Thus the overall yield of signal D+→ K−K+π+candidates in the three samples within
the mass window is approximately 370,000. The total number of candidates (S + B) in each decay mode used in the analysis are given in Table II. The Dalitz plot of data in the D+ window is shown in Fig. 2.
Within the 2σ D+ → K−K+π+ mass window, about
8.6% of events are background. Apart from random three-body track combinations, charm backgrounds and two-body resonances plus one track are expected. Charm reflections appear when a particle is wrongly identified in a true charm three-body decay and/or a track in a four-body charm decay is lost. The main three-four-body reflec-tion in the K−K+π+ spectrum is the Cabibbo-favoured
TABLE I. Yield (S) and purity for samples 1 and 3 after
the final selection. The purity is estimated in the 2σ mass window.
Decay Yield Purity
Sample 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 the subsequent analysis.
sample 1 sample 2 sample 3 Total
D+→ K−
K+π+ 84,667 65,781 253,446 403,894
D+s → K−K+π+ 126,206 91,664 346,068 563,938
D+→ K−
π+π+ 858,356 687,197 2,294,315 3,839,868
D+ → K−π+π+, where the incorrect assignment of the
kaon 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 lost
and the π− is misidentified as a K− will appear broadly distributed in K−K+π+ mass, but its resonances could
create structures in the Dalitz plot. Similarly, K∗(892)0
and φ resonances from the PV misreconstructed with a random track forming a three-body vertex will also ap-pear.
TABLE III. The CLEO-c amplitude model “B” [7] used in the simulation studies. The uncertainties are statistical, experimental systematic 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 verify that we can detect CPV with the strategy outlined in Sect. I without producing fake signals, and to devise and test suitable binning schemes for the Dalitz plot. They are also used to quantify our sensitivity to possible man-ifestations of CPV, where we define the sensitivity to a given level of CPV as the probability of observing it with 3σ significance.
For the D+→ K−K+π+Dalitz plot model, the result
of the CLEO-c analysis (fit B) [7] is used. The ampli-tudes and phases of the resonances used in this model are reproduced in Table III. For simplicity, only resonant modes with fit fractions greater than 2% are included in the pseudo-experiments. The fit fraction for a reso-nance is defined as the integral of its squared amplitude over the Dalitz plot divided by the integral of the square of the overall complex amplitude. An efficiency func-tion is determined from a two-dimensional second order polynomial fit to the Dalitz plot distribution of triggered events that survive the selection cuts in the GEANT-based [13] LHCb Monte Carlo simulation for nonresonant D+ → K−K+π+. A simple model for the background
is inferred from the Dalitz plots of the sidebands of the D+→ K−K+π+signal. It is composed of random
com-binations of K−, K+, and π+ tracks, φ resonances with
π+ 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 to limited numbers of MC events), so the method is tested on a range of different Dalitz plot models.
Pseudo-experiments with large numbers of events are used to investigate how CPV would be observed in the Dalitz plot. These experiments are simple “toy” simu-lations that produce points in the Dalitz plot according to the probability density function determined from the CLEO-c amplitude model with no representation of the proton-proton collision, detector, or trigger. Figure 3(a) illustrates the values of Si
CP observed with 8 × 107events
and no CPV. This dataset is approximately 50 times larger than the data sample under study. The result-ing χ2/ndf is 253.4/218, giving a p-value for consistency with 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 events with a CP violating phase difference of 4◦ between the amplitudes for the φ(1020)π+component in D+and D−
decays. The p-value in this case is less than 10−100. The CPV effect is clearly visible, and is spread over a broad area of the plot, changing sign from left to right. This sign change means the CPV causes only a 0.1% differ-ence in the total decay rate between D+ and D−. This
illustrates 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 with no systematic uncertainties, to see a 0.1% asymmetry at the 3σ level would require 2.25 × 106 events. With the
method and much smaller dataset used here we would observe this signal at the 3σ level with 76% probability, as shown in Table IV below.
The sensitivity to a particular manifestation of CPV depends on the choice of binning. The fact that the CP -violating region in most of the pseudo-experiments covers a broad area of the Dalitz plot suggests that the optimal number of bins for this type of asymmetry is low. Each bin 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 part of a large bin has any CPV in it. To avoid effects due to excessive fluctuations, bins that contain fewer than 50 candidates are not used anywhere in the analysis. Such bins 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 of approximately equal population without separating D+
and D−. Two binnings that are found to have good sensitivity to the simulated asymmetries contain 25 bins (“Adaptive I”) arranged as shown in Fig. 4(a), and 106 bins (“Adaptive II”) arranged as shown in Fig. 4(b). For Adaptive I, a simulation of the relative value of the strong phase across the Dalitz plot in the CLEO-c amplitude model is used to refine the results of the algorithm: if the strong phase varies significantly across a bin, CP asymmetries are more likely to change sign. Therefore the bin boundaries are adjusted to minimise changes in
) 4 /c 2 (GeV 2 + π -K m 0.5 1 1.5 2 ) 4 /c 2 (GeV 2 + K -K m 1 1.5 2 2.5 3 CP S -3 -2 -1 0 1 2 3
LHCb
(a)
) 4 /c 2 (GeV 2 + π -K m 0.5 1 1.5 2 ) 4 /c 2 (GeV 2 + K -K m 1 1.5 2 2.5 3 CP S -15 -10 -5 0 5 10 15LHCb
(b)
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).
) 4 /c 2 (GeV 2 + π -K m 0.5 1 1.5 2 ) 4 /c 2 (GeV 2 + K -K m 1 1.5 2 2.5 3 1 10 2 10 3 10
LHCb
(a)
) 4 /c 2 (GeV 2 + π -K m 0.5 1 1.5 2 ) 4 /c 2 (GeV 2 + K -K m 1 1.5 2 2.5 3 1 10 2 10 3 10LHCb
(b)
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 of this simulation could, in principle, influence the binning and 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 “Uniform II”, are defined. These use regular arrays of rectangular bins of equal size.
The adaptive binnings are used to determine the sen-sitivity to several manifestations of CPV. With 200 test experiments of approximately the same size as the signal sample in data, including no asymmetries, no CP -violating signals are observed at the 3σ level with Adap-tive I or AdapAdap-tive II. The expectation is 0.3.
With the chosen binnings, a number of sets of 100 pseudo-experiments with different CP -violating asymme-tries are produced. The probability of observing a given signal in either the φ(1020) or κ(800) resonances with 3σ significance is calculated in samples of the same size as the dataset. The results are given in Table IV. The CPV 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 be at least 95% confident of detecting the central values of these asymmetries.
The sensitivity of the results to variations in the Dalitz plot model and the background is investigated, and ex-ample results for the CP asymmetry in the φ(1020) phase are shown in Table V. In this table, models A and B are taken from the CLEO paper, model B2 includes an f0(980) contribution that accounts for approximately 8%
2The conventions used in the CLEO paper to define asymmetry are different, so the asymmetries in Table II of [7] have been multiplied by two in order to be comparable with those given above.
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. hSi is the mean significance with which CPV is observed.
CPV Adaptive I Adaptive II p(3σ) hSi p(3σ) hSi no CPV 0 0.84σ 1% 0.84σ 6◦in φ(1020) phase 99% 7.0σ 98% 5.2σ 5◦in φ(1020) phase 97% 5.5σ 79% 3.8σ 4◦in φ(1020) phase 76% 3.8σ 41% 2.7σ 3◦in φ(1020) phase 38% 2.8σ 12% 1.9σ 2◦in φ(1020) phase 5% 1.6σ 2% 1.2σ 6.3% in κ(800) magnitude 16% 1.9σ 24% 2.2σ 11% 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. hSi is the mean significance with which CPV is observed. The sample size is comparable to that seen in data.
Model Adaptive I Adaptive II
p(3σ) hSi p(3σ) hSi B (baseline) 76% 3.8σ 41% 2.7σ A 84% 4.3σ 47% 2.9σ B2 (add f0(980)) 53% 3.2σ 24% 2.2σ B3 (vary K∗0(1430)0 magn.) 82% 4.0σ 41% 2.8σ B4 (vary K∗0(1430)0 phase) 73% 3.7σ 38% 2.7σ
of events, and models B3 and B4 are variations of the K∗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 on the details of the model. This provides further justifi-cation for our model-independent approach. However, a reasonable level of sensitivity is retained in all the cases we tested. Thus, when taken together, the studies show that 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 occur precisely in the way investigated here.
IV. CONTROL MODES
It is possible that asymmetries exist in the data that do not result from CPV, for example due to production, backgrounds, instrumental effects such as left-right dif-ferences in detection efficiency, or momentum-dependent differences in the interaction cross-sections of the daugh-ter particles with detector madaugh-terial. Our sensitivity to such asymmetries is investigated in the two Cabibbo favoured control channels, where there is no large CPV predicted. The D+ → K−π+π+ control mode has an
order 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+ and K− reconstruction efficiencies. Conversely, the D+
s →
K−K+π+control mode is very similar to our signal mode
in terms of resonant structure, number of candidates, kinematics, detector effects, and backgrounds.
The control modes and their mass sidebands defined in Fig. 1 are tested for asymmetries using the method described in the previous section. Adaptive and uniform binning schemes are defined for D+ → K−π+π+ and
D+
s → K−K+π+. They are applied to samples 1–3 and
each magnet polarity separately. In the final results, the asymmetries measured in data taken with positive and negative magnet polarity are combined in order to cancel left-right detector asymmetries. The precise number of bins chosen is arbitrary, but care is taken to use a wide range of tests with binnings that reflect the size of the dataset for the decay mode under study.
For D+→ K−π+π+, five different sets of bins in each
scheme are used. A very low p-value would indicate a local asymmetry. One test with 25 adaptive bins in one of the subsamples (with negative magnet polarity) has a p-value of 0.1%, but when combined with the positive polarity sample the p-value increases to 1.7%. All other tests yield p-values ranging from 1–98%. Some example results are given in Table VI. A typical distribution of the SCP values with a Gaussian fit is shown in Fig. 5(a)
for a test with 900 uniform bins. The fitted values of the mean and width are consistent with one and zero respectively, suggesting that the differences between the D+and the D−Dalitz plots are driven only by statistical fluctuations.
CP S -4 -2 0 2 4 Number of bins (0.4) 0 20 40 60 80 100 120 140
LHCb
(a)
CP S -4 -2 0 2 4 Number of bins (0.5) 0 5 10 15 20 25 30LHCb
(b)
FIG. 5. (a) Distribution of SCP values from D+ → K−π+π+ from a test with 900 uniform bins. The mean of the fitted
Gaussian distribution is 0.015 ± 0.034 and the width is 0.996 ± 0.023. (b) Distribution of SCP values from Ds+→ K
−
K+π+
with 129 bins. The fitted mean is −0.011 ± 0.084 and the width is 0.958 ± 0.060.
) 4 /c 2 (GeV 2 , min + π -K m 1 1.5 2 2.5 3 ) 4 /c 2 (GeV 2 , max + π -K m 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 500 1000 1500 2000 2500 3000 3500 4000
LHCb
(a)
-0.45 0.66 -0.46 -0.96 -1.13 -1.71 0.70 0.64 0.70 0.05 -1.09 0.48 -1.74 0.49 2.62 -0.24 -0.04 -2.33 0.12 1.00 -1.38 0.31 1.02 0.69 1.80 ) 4 /c 2 (GeV 2 + π -K m 0.5 1 1.5 2 ) 4 /c 2 (GeV 2 + K -K m 1 1.5 2 2.5 3 3.5 1 10 2 10 3 10 4 10LHCb
(b)
Zone A Zone C Zone C Zone BFIG. 6. Dalitz plots of (a) D+→ K−
π+π+, showing the 25-bin adaptive scheme with the SCPvalues, 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 the
D+→ 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 in Sect. II.
1300 bins 900 bins 400 bins 100 bins 25 bins
Uniform 73.8 17.7 72.6 54.6 1.7
Adaptive 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 high density of events along the φ and the K∗(892)0 bands.
The Dalitz plot is divided into three zones, as shown in Fig. 6. Each zone is further divided into 300, 100 and 30 bins of same size. The results are given in Table VII. In addition, a test is performed on the whole Dalitz plot
us-TABLE VII. Results (p-values, in %) from tests with the
D+s → K−K+π+ control channel using the uniform binning
scheme. The values correspond to tests performed separately on Zones A-C, with samples 1-3 and both magnet polarities combined.
bins Zone A Zone B Zone C
300 20.1 25.3 14.5
100 41.7 84.6 89.5
30 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 by the ratio of the available phase space in the two modes. These tests yield p-values of 71.5% and 34.3% respec-tively.
TABLE VIII. Results (p-values, in %) from tests with the
D+ → K−π+π+ and D+s → K
−
K+π+ samples divided
ac-cording to the impact parameter with respect to the primary vertex. The tests are performed using the adaptive binning scheme with 25 bins.
χ2IP< 6 χ2IP > 6 D+→ K− π+π+ 8.5 88.9 D+s → K − K+π+ 52.0 30.6
signal region are the charm contamination of the back-ground, and asymmetries from CPV in misreconstructed B decays. In order to investigate the first possibility, similar tests are carried out in the mass sidebands of the D(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 charm is the impact parameter (IP) of the D with respect to the primary vertex. Given the long B lifetime, the IP distri-bution of secondary charm candidates is shifted towards larger values compared to that of prompt D+ mesons.
The effect of secondary charm is investigated by divid-ing the data set accorddivid-ing to the value of the candidate IP significance (χ2
IP). The subsample with events having
larger χ2
IP are likely to be richer in secondary charm. The
results are shown in Table VIII. No anomalous effects are seen in the high χ2IP sample, so contamination from sec-ondary charm with CPV does not affect our results for studies with our current level of sensitivity.
The analysis on the two control modes and on the sidebands in the final states K−K+π+ and K−π+π+ gives results from all tests that are fully consistent with no asymmetry. Therefore, any asymmetry observed in D+→ 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 the K−K+π+ mass window from 1856.7 to 1882.1 MeV/c2,
as described in Sect. II. There are 200,336 and 203,558 D+ 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 local asymmetries in the control channels D+→ K−π+π+and
D+
s → K−K+π+ and in the sidebands of the K−K+π+
mass spectrum, we investigate the signal sample under different binning choices.
First, the adaptive binning is used with 25 and 106 bins in the Dalitz plot as illustrated in Fig. 4. Then
CPV is investigated with uniform binnings, using 200 and 530 bins of equal size. For each of these binning choices, the significance Si
CP of the difference in D+and
D− population is computed for each bin i, as defined in Eq. 1. The χ2/ndf = P
i(S i CP)
2/ndf is calculated and
the p-value is obtained. The distributions of SCPi are fitted to Gaussian functions.
The p-values are shown in Table IX. The Dalitz plot distributions of SCPi are shown in Fig. 7. In Fig. 8 the dis-tributions of SCPi and the corresponding Gaussian fits for the different binnings are shown. The p-values obtained indicate no evidence for CPV. This is corroborated by the good fits of the Si
CP distributions to Gaussians, with
means 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 to 530 respectively. The samples are separated according to the magnet polarity and the same studies are re-peated. In all cases the p-values are consistent with no CPV, with values ranging from 4% to 99%. We conclude that there is no evidence for CPV in our data sample of D+→ K−K+π+.
VI. CONCLUSION
Due to the rich structure of their Dalitz plots, three body charm decays are sensitive to CP violating phases within and beyond the Standard Model. Here, a model-independent search for direct CP violation is performed in the Cabibbo suppressed decay D+→ K−K+π+ with
35 pb−1 of data collected by the LHCb experiment, and no evidence for CPV is found. Several binnings are used to compare normalised D+ and D− Dalitz plot
distri-butions. This technique is validated with large num-bers of simulated pseudo-experiments and with Cabibbo favoured control channels from the data: no false positive signals are seen. To our knowledge this is the first time a search for CPV is performed using adaptive bins which reflect the structure of the Dalitz plot.
Monte Carlo simulations illustrate that large localised asymmetries can occur without causing detectable differ-ences in integrated decay rates. The technique used here is shown to be sensitive to such asymmetries. Assuming the 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 the
κ(800)K+with 3σ significance. Since we find no evidence
of CPV, effects of this size are unlikely to exist.
VII. ACKNOWLEDGMENTS
We express our gratitude to our colleagues in the CERN accelerator departments for the excellent per-formance of the LHC. We thank the technical and
ad-TABLE IX. Fitted means and widths, χ2/ndf and p-values for consistency with no CPV for the D+→ K−
K+π+decay mode
with 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 12.7 Adaptive II −0.024 ± 0.010 1.078 ± 0.074 123.4/105 10.6 Uniform I −0.043 ± 0.073 0.929 ± 0.051 191.3/198 82.1 Uniform II −0.039 ± 0.045 1.011 ± 0.034 519.5/529 60.5 ) 4 /c 2 (GeV 2 + π -K m 0.5 1 1.5 2 ) 4 /c 2 (GeV 2 + K -K m 1 1.5 2 2.5 3 CP S -3 -2 -1 0 1 2 3
LHCb
(a)
) 4 /c 2 (GeV 2 + π -K m 0.5 1 1.5 2 ) 4 /c 2 (GeV 2 + K -K m 1 1.5 2 2.5 3 CP S -3 -2 -1 0 1 2 3LHCb
(b)
) 4 /c 2 (GeV 2 + π -K m 0.5 1 1.5 2 ) 4 /c 2 (GeV 2 + K -K m 1 1.5 2 2.5 3 CP S -3 -2 -1 0 1 2 3LHCb
(c)
) 4 /c 2 (GeV 2 + π -K m 0.5 1 1.5 2 ) 4 /c 2 (GeV 2 + K -K m 1 1.5 2 2.5 3 CP S -3 -2 -1 0 1 2 3LHCb
(d)
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); SNSF and SER (Switzerland); NAS Ukraine (Ukraine); STFC (United Kingdom); NSF (USA). We also acknowledge the support received from the ERC under FP7 and the Region Auvergne.
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