EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)

 LHCb-PAPER-2011-032 CERN-PH-EP-2011-206

Measurement of mixing and $C\\!P$ violation parameters in two-body charm
decays

The LHCb Collaboration 111Authors are listed on the following pages.

A study of mixing and indirect $C\\!P$ violation in $D^{0}$ mesons through the
determination of the parameters $y_{C\\!P}$ and $A_{\Gamma}$ is presented. The
parameter $y_{C\\!P}$ is the deviation from unity of the ratio of effective
lifetimes measured in $D^{0}$ decays to the $C\\!P$ eigenstate $K^{+}K^{-}$
with respect to decays to the Cabibbo favoured mode $K^{-}\pi^{+}$. The result
measured using data collected by LHCb in 2010, corresponding to an integrated
luminosity of $29\mbox{\,pb}^{-1}$, is

$y_{C\\!P}=(5.5\pm 6.3_{\rm stat}\pm 4.1_{\rm syst})\times 10^{-3}.$

The parameter $A_{\Gamma}$ is the asymmetry of effective lifetimes measured in
decays of $D^{0}$ and $\kern 1.99997pt\overline{\kern-1.99997ptD}{}^{0}$
mesons to $K^{+}K^{-}$. The result is

$A_{\Gamma}=(-5.9\pm 5.9_{\rm stat}\pm 2.1_{\rm syst})\times 10^{-3}.$

A data-driven technique is used to correct for lifetime-biasing effects.

Submitted to JHEP

## The LHCb collaboration

R. Aaij23, C. Abellan Beteta35,n, 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, S. Belogurov30, 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, T. Bird50, A. Bizzeti17,h, P.M. Bjørnstad50, T. Blake37, F. Blanc38,
C. Blanks49, J. Blouw11, S. Blusk52, A. Bobrov33, V. Bocci22, A. Bondar33, N.
Bondar29, W. Bonivento15, S. Borghi47,50, A. Borgia52, T.J.V. Bowcock48, C.
Bozzi16, T. Brambach9, J. van den Brand24, J. Bressieux38, D. Brett50, M.
Britsch10, T. Britton52, N.H. Brook42, H. Brown48, A. Büchler-Germann39, I.
Burducea28, A. Bursche39, J. Buytaert37, S. Cadeddu15, 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. Carson49, K. Carvalho
Akiba2, G. Casse48, M. Cattaneo37, Ch. Cauet9, 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, A. Comerma-Montells35, F. Constantin28, A.
Contu51, A. Cook42, M. Coombes42, G. Corti37, G.A. Cowan38, R. Currie46, C.
D’Ambrosio37, P. David8, P.N.Y. David23, 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, L. Del Buono8, C. Deplano15, D.
Derkach14,37, O. Deschamps5, F. Dettori24, J. Dickens43, H. Dijkstra37, P.
Diniz Batista1, F. Domingo Bonal35,n, S. Donleavy48, F. Dordei11, A. Dosil
Suárez36, D. Dossett44, A. Dovbnya40, F. Dupertuis38, R. Dzhelyadin34, A.
Dziurda25, S. Easo45, U. Egede49, V. Egorychev30, S. Eidelman33, D. van
Eijk23, F. Eisele11, S. Eisenhardt46, R. Ekelhof9, L. Eklund47, Ch.
Elsasser39, D. Elsby55, D. Esperante Pereira36, L. Estève43, A.
Falabella16,14,e, E. Fanchini20,j, C. Färber11, 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, D. Gascon35, C. Gaspar37, N. Gauvin38, M. Gersabeck37,
T. Gershon44,37, Ph. Ghez4, V. Gibson43, V.V. Gligorov37, C. Göbel54, D.
Golubkov30, A. Golutvin49,30,37, A. Gomes2, H. Gordon51, M. Grabalosa
Gándara35, R. Graciani Diaz35, L.A. Granado Cardoso37, E. Graugés35, G.
Graziani17, A. Grecu28, E. Greening51, 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,
T. Hartmann56, J. He7, V. Heijne23, K. Hennessy48, P. Henrard5, J.A. Hernando
Morata36, E. van Herwijnen37, E. Hicks48, 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, M. Kaballo9, S.
Kandybei40, M. Karacson37, T.M. Karbach9, J. Keaveney12, I.R. Kenyon55, U.
Kerzel37, T. Ketel24, A. Keune38, B. Khanji6, Y.M. Kim46, M. Knecht38, P.
Koppenburg23, A. Kozlinskiy23, L. Kravchuk32, K. Kreplin11, M. Kreps44, G.
Krocker11, P. Krokovny11, F. Kruse9, K. Kruzelecki37, M. Kucharczyk20,25,37,j,
T. Kvaratskheliya30,37, V.N. La Thi38, D. Lacarrere37, G. Lafferty50, A.
Lai15, D. Lambert46, R.W. Lambert24, E. Lanciotti37, G. Lanfranchi18, C.
Langenbruch11, T. Latham44, C. Lazzeroni55, R. Le Gac6, J. van Leerdam23,
J.-P. Lees4, R. Lefèvre5, A. Leflat31,37, J. Lefrançois7, O. Leroy6, T.
Lesiak25, L. Li3, L. Li Gioi5, M. Lieng9, M. Liles48, R. Lindner37, C. Linn11,
B. Liu3, G. Liu37, J. von Loeben20, J.H. Lopes2, E. Lopez Asamar35, N. Lopez-
March38, H. Lu38,3, J. Luisier38, A. Mac Raighne47, 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.
Märki38, J. Marks11, G. Martellotti22, A. Martens8, L. Martin51, A. Martín
Sánchez7, D. Martinez Santos37, A. Massafferri1, Z. Mathe12, C. Matteuzzi20,
M. Matveev29, E. Maurice6, B. Maynard52, A. Mazurov16,32,37, G. McGregor50, R.
McNulty12, 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.
Müller39, R. Muresan28,38, B. Muryn26, B. Muster38, M. Musy35, J. Mylroie-
Smith48, P. Naik42, T. Nakada38, R. Nandakumar45, I. Nasteva1, M. Nedos9, M.
Needham46, N. Neufeld37, C. Nguyen-Mau38,o, M. Nicol7, V. Niess5, N.
Nikitin31, A. Nomerotski51, A. Novoselov34, A. Oblakowska-Mucha26, V.
Obraztsov34, S. Oggero23, S. Ogilvy47, O. Okhrimenko41, R. Oldeman15,d, M.
Orlandea28, J.M. Otalora Goicochea2, P. Owen49, K. Pal52, J. Palacios39, A.
Palano13,b, M. Palutan18, J. Panman37, A. Papanestis45, M. Pappagallo47, C.
Parkes50,37, 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. Pérez-Calero Yzquierdo35,
P. Perret5, M. Perrin-Terrin6, G. Pessina20, A. Petrella16,37, A.
Petrolini19,i, A. Phan52, E. Picatoste Olloqui35, B. Pie Valls35, B.
Pietrzyk4, T. Pilař44, D. Pinci22, R. Plackett47, S. Playfer46, M. Plo
Casasus36, G. Polok25, A. Poluektov44,33, E. Polycarpo2, D. Popov10, B.
Popovici28, C. Potterat35, A. Powell51, 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. Rodrigues47,50,
F. Rodrigues2, P. Rodriguez Perez36, G.J. Rogers43, S. Roiser37, V.
Romanovsky34, M. Rosello35,n, 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. Schiller24, S.
Schleich9, M. Schlupp9, M. Schmelling10, B. Schmidt37, O. Schneider38, A.
Schopper37, M.-H. Schune7, R. Schwemmer37, B. Sciascia18, A. Sciubba18,l, M.
Seco36, A. Semennikov30, K. Senderowska26, I. Sepp49, N. Serra39, J. Serrano6,
P. Seyfert11, M. Shapkin34, I. Shapoval40,37, P. Shatalov30, Y. Shcheglov29,
T. Shears48, L. Shekhtman33, O. Shevchenko40, V. Shevchenko30, A. Shires49, R.
Silva Coutinho44, T. Skwarnicki52, A.C. Smith37, N.A. Smith48, E. Smith51,45,
K. Sobczak5, F.J.P. Soler47, A. Solomin42, F. Soomro18, 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,
V.K. Subbiah37, S. Swientek9, M. Szczekowski27, P. Szczypka38, T. Szumlak26,
S. T’Jampens4, E. Teodorescu28, F. Teubert37, C. Thomas51, E. Thomas37, J. van
Tilburg11, V. Tisserand4, M. Tobin39, S. Topp-Joergensen51, N. Torr51, E.
Tournefier4,49, 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, B. Viaud7, I. Videau7, X. Vilasis-Cardona35,n, J. Visniakov36, A.
Vollhardt39, D. Volyanskyy10, D. Voong42, A. Vorobyev29, H. Voss10, S.
Wandernoth11, J. Wang52, D.R. Ward43, N.K. Watson55, 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, W.
Witzeling37, S.A. Wotton43, K. Wyllie37, Y. Xie46, F. Xing51, Z. Xing52, 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 Físicas (CBPF), Rio de Janeiro, Brazil
2Universidade Federal do Rio de Janeiro (UFRJ), Rio de Janeiro, Brazil
3Center for High Energy Physics, Tsinghua University, Beijing, China
4LAPP, Université de Savoie, CNRS/IN2P3, Annecy-Le-Vieux, France
5Clermont Université, Université Blaise Pascal, CNRS/IN2P3, LPC, Clermont-
Ferrand, France
6CPPM, Aix-Marseille Université, CNRS/IN2P3, Marseille, France
7LAL, Université Paris-Sud, CNRS/IN2P3, Orsay, France
8LPNHE, Université Pierre et Marie Curie, Université Paris Diderot,
CNRS/IN2P3, Paris, France
9Fakultät Physik, Technische Universität Dortmund, Dortmund, Germany
10Max-Planck-Institut für Kernphysik (MPIK), Heidelberg, Germany
11Physikalisches Institut, Ruprecht-Karls-Universität Heidelberg, Heidelberg,
Germany
12School of Physics, University College Dublin, Dublin, Ireland
13Sezione INFN di Bari, Bari, Italy
14Sezione INFN di Bologna, Bologna, Italy
15Sezione INFN di Cagliari, Cagliari, Italy
16Sezione INFN di Ferrara, Ferrara, Italy
17Sezione INFN di Firenze, Firenze, Italy
18Laboratori Nazionali dell’INFN di Frascati, Frascati, Italy
19Sezione INFN di Genova, Genova, Italy
20Sezione INFN di Milano Bicocca, Milano, Italy
21Sezione INFN di Roma Tor Vergata, Roma, Italy
22Sezione INFN di Roma La Sapienza, Roma, Italy
23Nikhef National Institute for Subatomic Physics, Amsterdam, The Netherlands
24Nikhef National Institute for Subatomic Physics and Vrije Universiteit,
Amsterdam, The Netherlands
25Henryk Niewodniczanski Institute of Nuclear Physics Polish Academy of
Sciences, Kraców, Poland
26AGH University of Science and Technology, Kraców, 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 Fédérale de Lausanne (EPFL), Lausanne, Switzerland
39Physik-Institut, Universität Zürich, Zürich, 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
54Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), Rio de
Janeiro, Brazil, associated to 2
55University of Birmingham, Birmingham, United Kingdom
56Physikalisches Institut, Universität Rostock, Rostock, Germany, associated
to 11

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

##  1 Introduction

Mixing of neutral $D^{0}$ mesons has only recently been established [1, 2, 3]
and first evidence for $C\\!P$ violation in the charm sector has just been
seen by LHCb [4]. In this work the mixing and $C\\!P$ violation parameters
$y_{C\\!P}$ and $A_{\Gamma}$ in the decays of neutral $D^{0}$ mesons into two
charged hadrons are studied. Both quantities are measured here for the first
time at a hadron collider. The observable $y_{C\\!P}$ is the deviation from
unity of the ratio of inverse effective lifetimes in the decay modes
$D^{0}\rightarrow K^{+}K^{-}$ and $D^{0}\rightarrow K^{-}\pi^{+}$

$y_{C\\!P}\equiv\frac{\hat{\Gamma}(D^{0}\rightarrow
K^{+}K^{-})}{\hat{\Gamma}(D^{0}\rightarrow K^{-}\pi^{+})}-1,$ (1)

where effective lifetime refers to the value measured using a single
exponential model. All decays implicitly include their charge conjugate modes,
unless explicitly stated otherwise. Similarly, $A_{\Gamma}$ is given by the
asymmetry of inverse effective lifetimes as

$A_{\Gamma}\equiv\frac{\hat{\Gamma}(D^{0}\rightarrow
K^{+}K^{-})-\hat{\Gamma}(\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}\rightarrow
K^{+}K^{-})}{\hat{\Gamma}(D^{0}\rightarrow K^{+}K^{-})+\hat{\Gamma}(\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}\rightarrow K^{+}K^{-})}.$ (2)

The neutral $D^{0}$ mass eigenstates $|D_{1,2}\rangle$ with masses $m_{1,2}$
and widths $\Gamma_{1,2}$ can be expressed as linear combinations of the
flavour eigenstates as $|D_{1,2}\rangle=p|D^{0}\rangle\pm{}q|\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}\rangle$ with complex coefficients
$p$ and $q$ satisfying $|p|^{2}+|q|^{2}=1$. The average mass and width are
defined as $m\equiv(m_{1}+m_{2})/2$ and
$\Gamma\equiv(\Gamma_{1}+\Gamma_{2})/2$; the mass and width difference are
used to define the mixing parameters $x\equiv(m_{2}-m_{1})/\Gamma$ and
$y\equiv(\Gamma_{2}-\Gamma_{1})/(2\Gamma$). The phase convention is chosen
such that $C\\!P|D^{0}\rangle=-|\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}\rangle$ and $C\\!P|\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}\rangle=-|D^{0}\rangle$ which leads,
in the case of no $C\\!P$ violation ($p=q$), to $|D_{1}\rangle$ being the
$C\\!P$ odd and $|D_{2}\rangle$ the $C\\!P$ even eigenstate, respectively.

The parameter

$\lambda_{f}=\frac{q\bar{A}_{f}}{pA_{f}}=-\eta_{C\\!P}\left|\frac{q}{p}\right|\left|\frac{\bar{A}_{f}}{A_{f}}\right|e^{i\phi},$
(3)

contains the amplitude $A_{f}$ ($\bar{A}_{f}$) of $D^{0}$ ($\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}$) decays to the $C\\!P$ eigenstate
$f$ with eigenvalue $\eta_{C\\!P}$. The mixing parameters $x$ and $y$ are
known to be at the level of $10^{-2}$ while both the phase and the deviation
of the magnitude from unity of $\lambda_{f}$ are experimentally only
constrained to about $0.2$ [5]. The direct $C\\!P$ violation, i.e. the
difference in the rates of $D^{0}$ and $\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}$ decays, is constrained to the
level of $10^{-2}$ and has recently been measured by LHCb [4]. Introducing
$|q/p|^{\pm 2}\approx 1\pm\mathrm{A_{\rm m}}$ and $|\bar{A}_{f}/A_{f}|^{\pm
2}\approx 1\pm\mathrm{A_{\rm d}}$, with the assumption that $\mathrm{A_{\rm
m}}$ and $\mathrm{A_{\rm d}}$ are small, and neglecting terms below $10^{-4}$
according to the experimental constraints, one obtains according to Ref. [6]

$y_{C\\!P}\approx\left(1+\frac{1}{8}\mathrm{A_{\rm
m}}^{2}\right)y\cos\phi-\frac{1}{2}\mathrm{A_{\rm m}}x\sin\phi.$ (4)

In the limit of no $C\\!P$ violation $y_{C\\!P}$ is equal to $y$ and hence
becomes a pure mixing parameter. However, once precise measurements of $y$ and
$y_{C\\!P}$ are made, any difference between $y$ and $y_{C\\!P}$ would be a
sign of $C\\!P$ violation.

Previous measurements of $y_{C\\!P}$ have been performed by BaBar and Belle.
The results are $y_{C\\!P}=(11.6\pm 2.2\pm 1.8)\times 10^{-3}$ [7] for BaBar
and $y_{C\\!P}=(13.1\pm 3.2\pm 2.5)\times 10^{-3}$ [2] for Belle. They are
consistent with the world average of $y=(7.5\pm 1.2)\times 10^{-3}$ [5].

The study of the lifetime asymmetry of $D^{0}$ and $\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}$ mesons decaying into the singly
Cabibbo-suppressed final state $K^{+}K^{-}$ can reveal indirect $C\\!P$
violation in the charm sector. The measurement can be expressed in terms of
the quantity $A_{\Gamma}$. Using the same expansion as for $y_{C\\!P}$ leads
to

$\displaystyle A_{\Gamma}$ $\displaystyle\approx$
$\displaystyle\bigg{[}\frac{1}{2}(\mathrm{A_{\rm m}}+\mathrm{A_{\rm
d}})y\cos\phi-x\sin\phi\bigg{]}\frac{1}{1+y_{C\\!P}}$ (5)
$\displaystyle\approx$ $\displaystyle\frac{1}{2}(\mathrm{A_{\rm
m}}+\mathrm{A_{\rm d}})y\cos\phi-x\sin\phi.$

Despite this measurement being described in most literature as a determination
of indirect $C\\!P$ violation it is apparent that direct $C\\!P$ violation at
the level of $10^{-2}$ can have a contribution to $A_{\Gamma}$ at the level of
$10^{-4}$. Therefore precise measurements of both time-dependent and time-
integrated asymmetries are necessary to reveal the nature of $C\\!P$ violating
effects in the $D^{0}$ system.

The measurement of $A_{\Gamma}$ requires tagging the flavour of the $D^{0}$ at
production, which will be discussed in the following section. Previous
measurements of $A_{\Gamma}$ were performed by Belle and BaBar leading to
$A_{\Gamma}=(0.1\pm 3.0\pm 1.5)\times 10^{-3}$ [2] and $A_{\Gamma}=(2.6\pm
3.6\pm 0.8)\times 10^{-3}$ [8], respectively. They are consistent with zero,
hence showing no indication of $C\\!P$ violation.

##  2 Data selection

LHCb is a precision heavy flavour experiment which exploits the abundance of
charm particles produced in collisions at the Large Hadron Collider (LHC). The
LHCb detector [9] is a single arm spectrometer at the LHC with a
pseudorapidity acceptance of $2<\eta<5$ for charged particles. High precision
measurements of flight distances are provided by the Vertex Locator (VELO),
which consists of two halves with a series of semi-circular silicon microstrip
detectors. The VELO measurements, together with momentum information from
forward tracking stations and a $4~{}\mathrm{Tm}$ dipole magnet, lead to
decay-time resolutions of the order of one tenth of the $D^{0}$ lifetime. Two
Ring-Imaging Cherenkov (RICH) detectors using three different radiators
provide excellent pion-kaon separation over the full momentum range of
interest. The detector is completed by hadronic and electromagnetic
calorimeters and muon stations. The measurements presented here are based on a
data sample corresponding to an integrated luminosity of $29\mbox{\,pb}^{-1}$
of $pp$ collisions at $\sqrt{s}=7\mathrm{\,Te\kern-1.00006ptV}$ recorded
during the LHC run in 2010.

###  2.1 Trigger selection

The LHCb trigger consists of hardware and software (HLT) stages. The hardware
trigger is responsible for reducing the LHC $pp$ interaction rate from $\cal
O$(10) MHz to the rate at which the LHCb subdetectors can be read out,
nominally 1 MHz. It selects events based on the transverse momentum of track
segments in the muon stations, the transverse energy of clusters in the
calorimeters, and overall event multiplicity.

The HLT further reduced the event rate to about $2~{}{\rm\,kHz}$ in 2010, at
which the data was stored for offline processing. The HLT runs the same
software for the track reconstruction and event selection as is used offline
and has access to the full event information.

The first part of the HLT is based on the reconstruction of tracks and primary
interaction vertices in the VELO. Heavy flavour decays are identified by their
large lifetimes, which cause their daughter tracks to be displaced from the
primary interaction. The trigger first selects VELO tracks whose distance of
closest approach to any primary interaction, known as the impact parameter
(IP), exceeds $110~{}\,\upmu\rm m$. In addition the tracks are required to
have at least ten hits in the VELO to reduce further the accepted rate of
events. This cut limits the fiducial volume for $D^{0}$ decays and therefore
rejects events where the $D^{0}$ candidate has a large transverse component of
the distance of flight, causing an upper bound on the decay-time acceptance.
The term decay-time acceptance will be used throughout this paper to refer to
the selection efficiency as a function of the $D^{0}$ decay time. Selected
tracks are then used to define a region of interest in the tracking stations
after the dipole magnet, whose size is defined by an assumed minimum track
momentum of $8{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$; hits inside these search
regions are used to form tracks traversing the full tracking system. Tracks
passing this selection are fitted, yielding a full covariance matrix, and a
final selection is made based on the track-fit quality and the track
$\chi^{2}({\rm IP})$. The $\chi^{2}({\rm IP})$ is a measure of the consistency
with the hypothesis that the IP is equal to zero. At least one good track is
required for the event to be accepted. The requirements on both the track IP
and on the $\chi^{2}({\rm IP})$ reduce the number of $D^{0}$ candidates with a
short decay time.

In the second part of the HLT, an exclusive selection of $D^{0}$ candidates is
performed by reconstructing two-track vertices. Further cuts are placed on the
$\chi^{2}({\rm IP})$ of the $D^{0}$ daughters and the displacement
significance of the $D^{0}$ vertex from the primary interaction, as well as a
requirement which limits the collinearity angle between the $D^{0}$ momentum
and the direction of flight, as defined by the primary and decay vertices.
These cuts all affect the distribution of the decay time of the $D^{0}$
candidates. Additional cuts are placed on track and vertex fit quality, and on
kinematic quantities such as the transverse momentum of the $D^{0}$ candidate,
which have no effect on the decay-time distribution.

###  2.2 Offline selection

Given the abundance of charm decays, the selection has been designed to
achieve high purity. It uses similar requirements to those made in the trigger
selection, though often with tighter thresholds. In addition it makes use of
the RICH information for separating kaons and pions to achieve a low
misidentification rate. A mass window of
$\pm{}16{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ (about $\pm 2\sigma$) is
applied to the invariant mass of the two $D^{0}$ daughter particles using the
appropriate mass hypotheses. After these criteria have been applied there is
negligible remaining cross-feed between the different two-body $D^{0}$ decay
modes.

Flavour tagging of the $D^{0}$ decays is done by reconstructing the
$D^{*+}\rightarrow D^{0}\pi_{\rm s}^{+}$ decay, where the charge of the slow
pion, $\pi_{\rm s}$, determines the flavour of the $D^{0}$ meson at
production. The selection applies loose requirements on the kinematics of the
bachelor pion and the quality of the $D^{*+}$ vertex fit. The most powerful
variable for selecting the $D^{*+}$ decay is the difference in the
reconstructed invariant masses of the $D^{*+}$ and the $D^{0}$ candidates,
$\Delta m$. Candidates are required to have $\Delta m$ in the range $|\Delta
m-145.4{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}|<2.0{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$.

Events with multiple signal candidates are excluded from the analysis. For
tagged $D^{0}$ decays this causes a reduction of the number of candidates of
about $15\%$ due to the high probability of assigning a random slow pion to
form a $D^{*+}$ candidate. The numbers of selected candidates are 286,155 for
$D^{0}\\!\rightarrow K^{-}\pi^{+}$ and 39,262 for $D^{0}\\!\rightarrow
K^{+}K^{-}$ decays.

##  3 Determination of proper-time acceptance effects

Since absolute lifetime measurements are used to extract $y_{C\\!P}$ and
$A_{\Gamma}$, it is essential to correct for lifetime-biasing effects. The
analysis uses a data-driven approach that calculates, for each candidate and
at every possible decay time, an acceptance value of zero or one which is
related to the trigger decision and offline selection. The final per-event
acceptance function is used in the normalisation of the decay-time probability
density function (PDF) as described in the following section.

The method used to determine decay-time acceptance effects is based on the so-
called “swimming” algorithm. This approach was first used at the NA11
spectrometer [10], further developed within DELPHI [11] and CDF [12, 13],
studied at LHCb [14, 15], and applied to the measurement of the
$B^{0}_{s}\rightarrow K^{+}K^{-}$ lifetime [16].

The key to this method is the ability to execute the LHCb trigger software,
including the reconstruction, in precisely the same configuration used during
data taking. This is made possible by the implementation of all lifetime-
biasing requirements of the trigger in software rather than in the hardware.
The acceptance as a function of decay time is evaluated per event by
artificially moving the position of the primary interaction vertices
reconstructed in the trigger along the direction of the $D^{0}$ momentum in
order to give the $D^{0}$ candidate a different decay time. In events
containing multiple primary vertices, all are moved coherently in the
direction of the $D^{0}$ momentum. An analogous procedure is used to obtain
the decay-time acceptance of the offline selection.

A decay-time acceptance function for any single event is in the simplest case
a step function, as shown in Fig. 1, since the kinematics and chosen decay
time of the $D^{0}$ decay fully determine whether the event is triggered by
this $D^{0}$ candidate or not. It is important to note that the acceptance
function for a given event does not depend on the measured decay time of that
event, $t_{\rm meas}$. Accepted (rejected) regions take an acceptance value of
$1$ ($0$). In this method decay-time independent selection efficiencies are
factorised out and hence do not affect the result. The presence of additional
interaction vertices can lead to regions of no acceptance and the VELO
geometry puts an upper limit on the accepted range. Thus, a general decay-time
acceptance function is given as a series of steps or top-hat functions. The
decay times at which the event enters or leaves one of these top hats are
called turning points. The acceptance functions of the trigger and offline
selections are combined to a single acceptance function by including only the
ranges which have been accepted by both selections.

(a)

(b)

(c)

Figure 1: Variation of the decay-time acceptance function for a two-body
$D^{0}$ decay when moving the primary vertex along the $D^{0}$ momentum
vector. The shaded, light blue regions show the bands for accepting a track
impact parameter. While the impact parameter of the negative track (IP2) is
too low in (a) it reaches the accepted range in (b). The actual measured decay
time, $t_{\rm meas}$, lies in the accepted region which continues to larger
decay times (c).

The idea of studying the decay-time dependence of the acceptance in principle
requires moving the hits produced by the $D^{0}$ decay products. The
implementation of moving the primary vertices instead leads to significant
technical simplifications. However, this procedure ignores the fact that
events are no longer accepted if the mother particle has such a long decay
time that one or both tracks can no longer be reconstructed inside the VELO.
This is a very small effect as a $D^{0}$ meson has to fly ten to a hundred
times its average distance of flight in order to escape detection in the VELO.
Nevertheless, this effect can be estimated based on the knowledge of the
position of the VELO modules and on the number of hits required to form a
track. Using the information on the position of the VELO sensors, the limit of
the acceptance is determined by swimming the tracks along the $D^{0}$ momentum
vector. The result is treated as another per event decay-time acceptance and
merged with the swimming results of the trigger and offline selections.

Finally, the track reconstruction efficiency in the trigger is reduced
compared to the offline reconstruction due to the requirements described in
Sect. 2. It has been verified, using a smaller sample acquired without a
lifetime biasing selection, that this relative reconstruction efficiency does
not depend on the decay time of the $D^{0}$ candidate with a precision of
$3\times 10^{-3}$, and therefore introduces no significant additional
acceptance effect.

##  4 Fitting method

The peak in $\Delta m$ from true $D^{*+}$ decays is parametrised as the sum of
three Gaussians; two of which have a common mean and a third which has a
slightly higher mean. The random $\pi_{\rm s}$ background PDF is given by

$f_{\pi_{\rm s}}(\Delta m)=\left(\frac{\Delta
m}{a}\right)^{2}\>\left(1-\exp(-\frac{\Delta
m-d}{c})\right)\>+\>b\left(\frac{\Delta m}{d}-1\right)\qquad\Delta m\geq d,$
(6)

where $a$ and $b$ define the slope at high values of $\Delta m$, $c$ defines
the curvature at low values of $\Delta m$ and $\Delta m=d$ defines the
threshold below which the function is equal to zero. Figure 2 shows the
$\Delta m$ vs $m_{D^{0}}$ distribution and Fig. 3 shows the fit to the mass
difference between the reconstructed invariant masses of $D^{*+}$ and $D^{0}$
candidates, $\Delta m$.

Figure 2: $\Delta m$ vs $m_{D^{0}}$ distribution for $D^{0}\rightarrow
K^{-}\pi^{+}$ candidates. The contribution of random slow pions extends around
the signal peak in the vertical direction while background is visible as a
horizontal band.

Figure 3: $\Delta m$ fit projections of (left) $D^{0}\rightarrow K^{-}\pi^{+}$
and (right) $D^{0}\rightarrow K^{+}K^{-}$ candidates. Shown are data (points),
the total fit (green, solid) and the background component (blue, dot-dashed).

The signal yield is extracted from fits to the reconstructed $D^{0}$ invariant
mass distribution after application of the cut in $\Delta m$. The fit model
for the signal peak has been chosen to be a double Gaussian and background is
modelled as a first-order polynomial. The background level is evaluated to be
about $1\%$ for $D^{0}\\!\rightarrow K^{-}\pi^{+}$ decays and about $3\%$ for
$D^{0}\\!\rightarrow K^{+}K^{-}$ decays. It consists of combinatorial
background and partially reconstructed or misidentified $D^{0}$ decays. If the
latter stem from a $D^{*+}$ decay they have a peaking distribution in $\Delta
m$ similar to signal candidates. The data in the mass sidebands are
insufficient to reliably describe the background shape in other variables, so
the background contribution is neglected in the time-dependent fit and a
systematic uncertainty is estimated accordingly.

Events inside the signal windows in $\Delta m$ and $m_{D^{0}}$ are used in the
lifetime fit, where $D^{0}$ mesons produced at the primary vertex (prompt)
have to be distinguished from those originating from $b$ hadron decays
(secondary). The combined PDF for this decay-time dependent fit is factorized
as

$f(\chi^{2}({\rm IP}_{D}),t,A)=\sum_{\begin{subarray}{c}{\rm class}\\\ ={\rm
prompt},\\\ {\rm secondary}\end{subarray}}f_{\text{IP}}(\chi^{2}({\rm
IP}_{D})|t,A,{\rm class})\>f_{t}(t|A,{\rm class})\>f_{\text{TP}}(A|{\rm
class})\>P({\rm class}).$ (7)

The four factors on the right-hand side of Eq. 7, which will be described in
detail below, are:

  * •

the time-dependent PDFs for the $\ln\chi^{2}({\rm IP}_{D})$ values for prompt
and secondary $D^{0}$ mesons;

  * •

the decay-time PDFs for prompt and secondary $D^{0}$ mesons;

  * •

the PDF for the turning points which define the acceptance $A$;

  * •

the fractions of prompt and secondary $D^{0}$ decays among the signal
candidates.

The separation of prompt and secondary $D^{0}$ mesons is done on a statistical
basis using the impact parameter of the $D^{0}$ candidate with respect to the
primary vertex, ${\rm IP}_{D}$. For prompt decays, this is zero up to
resolution effects, but can acquire larger values for secondary decays as the
$D^{0}$ candidate does not in general point back to the primary vertex. Given
an estimate of the vertex resolution is available on an event-by-event basis,
it is advantageous to use the $\chi^{2}$ of the ${\rm IP}_{D}$ instead of the
impact parameter value itself. The natural logarithm of this quantity,
$\ln(\chi^{2}({\rm IP}_{D}))$, allows for an easier parametrisation.
Empirically, the sum of two bifurcated Gaussians, i.e. Gaussians with
different widths on each side of the mean, and a third, symmetric Gaussian,
all sharing a common peak position, is found to be a suitable model to
describe the $\ln(\chi^{2}({\rm IP}_{D}))$ distribution for both prompt and
secondary $D^{0}$.

For the prompt $D^{0}$ class the $\ln(\chi^{2}({\rm IP}_{D}))$ distribution
does not change with $D^{0}$ decay time as the true value is zero at all times
and the resolution of ${\rm IP}_{D}$ can be assumed to be independent of the
measured decay time. For secondary $D^{0}$ decays the decay-time and
$\ln(\chi^{2}({\rm IP}_{D}))$ are correlated. The width of the
$\ln(\chi^{2}({\rm IP}_{D}))$ distribution is found to be approximately
constant in decay time for both prompt and secondary $D^{0}$ mesons. As Monte
Carlo simulation studies suggest that secondary decays have a larger width in
this variable, a scale factor between the widths for prompt and secondary
mesons is introduced. The mean value of $\ln(\chi^{2}({\rm IP}_{D}))$
increases with $D^{0}$ decay time, which reflects the fact that $D^{0}$ mesons
coming from other long-lived decays do not necessarily point back to the
primary vertex and that they may point further away the further their parent
particle flies. The functional form for this time dependence is based on
simulation and all parameters are determined in the fit to data.

The decay-time PDF, $f_{t}(t|A,{\rm class})$ is modelled as a single
exponential for the prompt $D^{0}$ class and as a convolution of two
exponentials for secondary decays. To account for resolution effects, these
are convolved with a single Gaussian resolution function. The parameters of
the resolution model are obtained from a fit to the decay time distribution of
prompt ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ events. The resulting
dilution is equivalent to that of a single Gaussian with a width of
$50\rm\,fs$ [17]. The decay-time probability densities are properly normalized
by integrating their product with the acceptance function $A$, evaluated by
the swimming method, only over the decay-time intervals for which the event
would have been accepted. Hence, the acceptance turning points are used as
boundaries in the integration.

Finally, a PDF for the per-event acceptance function is needed. While the
first acceptance turning point, i.e. the one with the smallest decay time,
depends on the $D^{0}$ decay topology, the others are governed more by the
underlying event structure, e.g. the distribution of primary vertices. The
primary vertex distribution is independent of whether the $D^{0}$ candidate is
of prompt or secondary origin. Hence, the PDF can be approximated as
$f_{\text{TP}}(A|{\rm class})\approx f_{\text{TP}}(\text{TP}_{1}|{\rm
class})$, where $\text{TP}_{1}$ denotes the position of the first turning
point. The distribution for $f_{\text{TP}}(\text{TP}_{1}|{\rm prompt})$ is
obtained by applying a cut at $\ln\chi^{2}({\rm IP}_{D})<1$, thus selecting a
very pure sample of prompt decays. The distribution for
$f_{\text{TP}}(\text{TP}_{1}|{\rm secondary})$ is obtained from the
distribution of $\text{TP}_{1}$ weighted by the probability of each candidate
being of secondary decay origin.

An initial fit is performed using the full $\ln\chi^{2}({\rm IP}_{D})$
distribution and all parameters in the description of this term are then fixed
in the final fit. A cut is then applied requiring $\ln\chi^{2}({\rm
IP}_{D})<2$ in order to suppress the fractions of both background and
secondary candidates to less than a few percent. The final fit is performed on
this reduced sample. The effect of this procedure is estimated in the
systematic uncertainty evaluation.

##  5 Cross-checks and systematic uncertainties

The method for absolute lifetime measurements described in Sect. 4 comprises
three main parts whose accuracy and potential for biasing the measurement have
to be evaluated in detail:

  * •

the determination of the event-by-event decay-time acceptance;

  * •

the separation of prompt from secondary charm decays;

  * •

the estimation of the decay time distribution of combinatorial background.

Since the contribution of combinatorial background is ignored in the fit, it
is important to evaluate the corresponding systematic uncertainty.
Furthermore, several other parameters are used in the fit whose systematic
effects have to be evaluated, e.g. the description of the decay-time
resolution. It is generally expected that the systematic uncertainties in
$y_{C\\!P}$ are similar to or larger than those in $A_{\Gamma}$ as in
$y_{C\\!P}$ two different final states contribute to the measurement.

Several consistency checks are performed by splitting the dataset into
subsets. The stability is tested as a function of run period, $D^{0}$ momentum
and transverse momentum, and primary vertex multiplicity. No significant trend
is observed and therefore no systematic uncertainty assigned.

The fitting procedure is verified using simplified Monte Carlo simulation
studies. No indication of a bias is observed and the statistical uncertainties
are estimated accurately.

As an additional check, a control measurement is performed using the lifetime
asymmetry of $D^{0}$ and $\kern 1.99997pt\overline{\kern-1.99997ptD}{}^{0}$
decays to the Cabibbo favoured decay $D^{0}\\!\rightarrow K^{-}\pi^{+}$. The
result is in agreement with zero and the flavour-averaged $D^{0}$ lifetime is
found to be consistent with the world average. Detailed results are given in
Sect. 6. The fit results for $D^{0}\\!\rightarrow K^{+}K^{-}$ decays were not
looked at throughout the development of the method and the study of systematic
uncertainties for the analyses of $y_{C\\!P}$ and $A_{\Gamma}$.

###  5.1 Evaluation of systematic uncertainties

Particle decay times are measured from the distance between the primary vertex
and secondary decay vertex in the VELO. The systematic uncertainty from the
distance scale is determined by considering the potential error on the length
scale of the detector from the mechanical survey, thermal expansion and the
current alignment precision. A relative systematic uncertainty of $0.1\%$ is
assigned to the measurements of absolute lifetimes, translating into a
relative uncertainty of $0.1\%$ on $A_{\Gamma}$ and $y_{C\\!P}$.

The method to evaluate the turning points of the decay-time acceptance
functions described in Sect. 3 uses an iterative approach which estimates the
turning points to a precision of about $1\rm\,fs$. Two scenarios have been
tested: a common bias of all acceptance turning points and a common length
scaling of the turning points, which could originate from differences in the
length scale in the trigger and offline reconstructions. From a variation of
the bias and the scale, a systematic uncertainty of $0.1\times 10^{-3}$ on
$A_{\Gamma}$ and $y_{C\\!P}$ is determined.

The reconstruction acceptance is dominated by the VELO geometry, which is
accounted for by the method described in Sect. 3. This leads to a correction
of less than $1\rm\,fs$ on the absolute lifetime measurements, i.e. a relative
correction of about $0.24\%$. No further systematic uncertainty is assigned to
$A_{\Gamma}$ or $y_{C\\!P}$ as the size of this relative correction is
negligible. Additional studies of the reconstruction efficiency as a function
of variables governing the decay geometry did not provide any indication of
lifetime biasing effects.

The decay-time resolution is modelled by a single Gaussian. The width of the
resolution function is varied from its nominal value of $0.05{\rm\,ps}$
between $0.03{\rm\,ps}$ and $0.07{\rm\,ps}$. The range of variation was chosen
to cover possible alignment effects as well as effects from the different
final state used to evaluate the resolution. The result leads to a systematic
uncertainty of $0.1\times 10^{-3}$ for $A_{\Gamma}$ and $y_{C\\!P}$.

The fit range in decay time is restricted by lower and upper limits. The lower
limit is put in place to avoid instabilities in regions with extremely low
decay-time acceptances and very few events. The default cut value is
$0.25{\rm\,ps}$ which is close to the lower end of the observed range of
events. This cut is varied to both $0.2{\rm\,ps}$ and $0.3{\rm\,ps}$. The
result leads to a systematic uncertainty of $0.1\times 10^{-3}$ for
$A_{\Gamma}$ and $0.8\times 10^{-3}$ for $y_{C\\!P}$.

The upper limit of the fit range in decay time is put in place to minimise the
impact of long-lived background events. The default cut is put at $6{\rm\,ps}$
which corresponds to about $15$ $D^{0}$ lifetimes. This cut is varied to
$5{\rm\,ps}$ and $8{\rm\,ps}$. The result leads to a systematic uncertainty of
$0.2\times 10^{-3}$ for $A_{\Gamma}$ and $y_{C\\!P}$.

The description of the contribution from combinatorial background is studied
by varying its relative amount in the data sample and repeating the fit. This
is done by changing the $\Delta m$ window from the default of $\pm
2{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ to $\pm
1{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ and $\pm
3{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$. The result leads to a systematic
uncertainty of $1.3\times 10^{-3}$ for $A_{\Gamma}$ and $0.8\times 10^{-3}$
for $y_{C\\!P}$.

Events that originate from secondary charm decays are the background with the
largest impact on the fit procedure as they have a very different decay-time
distribution compared to prompt charm decays, but they peak in the invariant
mass and $\Delta m$ distributions. Also a fraction of combinatorial background
events appear to be secondary-like in their $\ln\chi^{2}({\rm IP}_{D})$
distribution. The cut of $\ln\chi^{2}({\rm IP}_{D})<2$ removes a large
fraction of secondary-like events. However, it is important that the remainder
is properly modelled and does not bias the signal lifetime. Varying this cut
changes the relative number of secondary-like decays in the sample and
therefore tests the stability of the secondary description in the fit model.
The fraction of secondary-like combinatorial background events is also altered
with this test. The $\ln\chi^{2}({\rm IP}_{D})$ cut is varied from $1.5$ which
is just above the peak of the prompt distribution to $3.5$ where the
probability densities for prompt and secondary decays are about equal. The
result leads to a systematic uncertainty of $1.6\times 10^{-3}$ for
$A_{\Gamma}$ and $3.9\times 10^{-3}$ for $y_{C\\!P}$. The uncertainty is
significantly larger for $y_{C\\!P}$ than for $A_{\Gamma}$ as may be expected
from the difference in the background level in the channels involved in the
$y_{C\\!P}$ measurement.

Additional studies were performed to estimate the potential impact of
neglecting background events in the fit. A background component was added to a
simplified simulation. The background decay time distribution was generated
using extreme values of fits to the distribution observed in mass sidebands.
The average bias on the measurement of $y_{C\\!P}$ was about $2\times
10^{-3}$. Since this is consistent with the assigned systematic uncertainty,
we do not assign any additional uncertainty.

Furthermore, a background component was added to the $D^{0}$ decay-time PDF
with a fixed fraction and average lifetime. The fraction of this component,
which was assumed to be secondary-like, was varied. A change in the fit result
for $y_{C\\!P}$ of $0$ (all background secondary-like) to $4\times 10^{-3}$
(all background prompt-like) was observed. As it is known that a fraction of
the background events are secondary-like, this result is considered consistent
with the simplified simulation results.

###  5.2 Summary of systematic uncertainties

Table 1 summarises the systematic uncertainties evaluated as described above.
The main systematic uncertainties are due to neglecting the combinatorial
background and to the contribution of secondary-like decays. The total
systematic uncertainties for $A_{\Gamma}$ and $y_{C\\!P}$, obtained by
combining all sources in quadrature, are $2.1\times 10^{-3}$ and $4.1\times
10^{-3}$, respectively.

Table 1: Summary of systematic uncertainties. Effect |  $A_{\Gamma}$ $(10^{-3})$ |  $y_{C\\!P}$ $(10^{-3})$
---|---|---
Decay-time acceptance correction | $0.1$ | $0.1$
Decay-time resolution | $0.1$ | $0.1$
Minimum decay-time cut | $0.1$ | $0.8$
Maximum decay-time cut | $0.2$ | $0.2$
Combinatorial background | $1.3$ | $0.8$
Secondary-like background | $1.6$ | $3.9$
Total | $2.1$ | $4.1$

##  6 Results and conclusion

The measurement of $y_{C\\!P}$ is based on absolute lifetime measurements as
described in Sect. 4. It uses flavour-tagged events reconstructed in the decay
chain $D^{*+}\rightarrow D^{0}\pi^{+}$, with $D^{0}$ and $\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}$ decays fitted simultaneously per
decay mode. The $\ln\chi^{2}({\rm IP}_{D})$ projection of the final fit is
shown in Fig. 4.

Figure 4: $\ln\chi^{2}({\rm IP}_{D})$ fit projection of $D^{0}\rightarrow
K^{+}K^{-}$ candidates in logarithmic scale. Shown are data (points), the
total fit (green, solid), the prompt signal (blue, short-dashed), and the
secondary signal (purple, long-dashed).

The result for the lifetime measured in $D^{0}\\!\rightarrow K^{-}\pi^{+}$
decays is $\tau(D^{0})=410.2\pm 0.9\rm\,fs$ where the uncertainty is
statistical only. The result for the lifetime is found to be in agreement with
the current world average [18]. Combining with the $D^{0}\\!\rightarrow
K^{+}K^{-}$ lifetime measurement, $\tau(D^{0})=408.0\pm 2.4_{\rm
stat}\rm\,fs$, this leads to the final result for $y_{C\\!P}$ of

$y_{C\\!P}=(5.5\pm 6.3_{\rm stat}\pm 4.1_{\rm syst})\times 10^{-3}.$

The measurement of $A_{\Gamma}$ is performed based on the same dataset and
applying the same fitting method as used for the extraction of $y_{C\\!P}$. A
control measurement is performed using decays to the Cabibbo favoured mode
$D^{0}\\!\rightarrow K^{-}\pi^{+}$ by forming a lifetime asymmetry analogous
to Eq. 2. The measured flavour-tagged lifetimes are effective parameters since
the fitted distributions also include mistagged events. For the control
measurement using $D^{0}\\!\rightarrow K^{-}\pi^{+}$ decays this contamination
is ignored as it is negligible due to the Cabibbo suppression of the mistagged
decays. The result for the asymmetry is
$A_{\Gamma}^{K\pi,\mathrm{eff}}=(-0.9\pm 2.2_{\rm stat})\times 10^{-3}$ which
is consistent with zero, according to expectations.

For the extraction of $A_{\Gamma}$, the mistagged decays are taken into
account by expressing the measured effective lifetimes, $\tau^{\mathrm{eff}}$,
in terms of the flavour-tagged lifetimes, $\tau(D^{0})$ and $\tau(\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0})$, and the mistag rate,
$\epsilon_{\pm}$, where the sign is according to the sign of the tagging pion:

$\displaystyle\tau^{\mathrm{eff}}(D^{0})$ $\displaystyle\approx$
$\displaystyle(1-\epsilon_{+})\>\tau(D^{0})+\epsilon_{+}\>\tau(\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0})$ (8)
$\displaystyle\tau^{\mathrm{eff}}(\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0})$ $\displaystyle\approx$
$\displaystyle(1-\epsilon_{-})\>\tau(\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0})+\epsilon_{-}\>\tau(D^{0}).$ (9)

The mistag rates are assumed to be independent of the final state and are
extracted from the favoured $D^{0}\\!\rightarrow K^{-}\pi^{+}$ decays as half
the fraction of the random slow pion background in the signal region of the
$\Delta m$ distribution. They are found to be about $1.8\%$. The systematic
uncertainty due to this correction is negligible.

Figure 5: Proper-time fit projections of (left) $D^{0}\rightarrow K^{+}K^{-}$
and (right) $\kern 1.99997pt\overline{\kern-1.99997ptD}{}^{0}\rightarrow
K^{+}K^{-}$ candidates after application of the $\ln\chi^{2}({\rm IP}_{D})<2$
cut. Shown are data (points), the total fit (green, solid), the prompt signal
(blue, short-dashed), and the secondary signal (purple, long-dashed).

The projection of the decay-time fit to $D^{0}$ and $\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}$ candidates in $D^{0}\\!\rightarrow
K^{+}K^{-}$ decays is shown in Fig. 5. After applying the mistag correction,
the resulting value of $A_{\Gamma}$ is

$A_{\Gamma}=(-5.9\pm 5.9_{\rm stat}\pm 2.1_{\rm syst})\times 10^{-3}.$

Both results on $y_{C\\!P}$ and $A_{\Gamma}$ are compatible with zero and in
agreement with previous measurements [2, 7, 8]. Future updates are expected to
lead to significant improvements in the sensitivity. The systematic
uncertainty is expected to be reduced by an improved treatment of background
events which will be possible for the data taken in 2011.

## Acknowledgements

We express our gratitude to our colleagues in the CERN accelerator departments
for the excellent performance of the LHC. We thank the technical and
administrative staff at CERN and at the LHCb institutes, and acknowledge
support from the National Agencies: 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 (The Netherlands); SCSR (Poland); ANCS
(Romania); MinES of Russia and Rosatom (Russia); 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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