Search for Violation of Lorentz Invariance in Top Quark Pair Production and Decay V.M. Abazov,32 B. Abbott,70 B. S. Acharya,26 M. Adams,46 T. Adams,44 G.D. Alexeev,32 G. Alkhazov,36 A. Alton,58,† G. Alverson,57 M. Aoki,45 A. Askew,44 S. Atkins,55 K. Augsten,7 C. Avila,5 F. Badaud,10 L. Bagby,45 B. Baldin,45 D.V. Bandurin,44 S. Banerjee,26 E. Barberis,57 P. Baringer,53 J. Barreto,2 J. F. Bartlett,45 U. Bassler,15 V. Bazterra,46 A. Bean,53 M. Begalli,2 L. Bellantoni,45 M. S. Berger,49 S. B. Beri,24 G. Bernardi,14 R. Bernhard,19 I. Bertram,39 M. Besançon,15 R. Beuselinck,40 V. A. Bezzubov,35 P. C. Bhat,45 S. Bhatia,60 V. Bhatnagar,24 G. Blazey,47 S. Blessing,44 K. Bloom,61 A. Boehnlein,45 D. Boline,67 E. E. Boos,34 G. Borissov,39 T. Bose,56 A. Brandt,73 O. Brandt,20 R. Brock,59 G. Brooijmans,65 A. Bross,45 D. Brown,14 J. Brown,14 X. B. Bu,45 M. Buehler,45 V. Buescher,21 V. Bunichev,34 S. Burdin,39,‡ C. P. Buszello,38 E. Camacho-Pérez,29 B. C.K. Casey,45 H. Castilla-Valdez,29 S. Caughron,59 S. Chakrabarti,67 D. Chakraborty,47 K.M. Chan,51 A. Chandra,75 E. Chapon,15 G. Chen,53 S. Chevalier-Théry,15 D. K. Cho,72 S.W. Cho,28 S. Choi,28 B. Choudhary,25 S. Cihangir,45 D. Claes,61 J. Clutter,53 M. Cooke,45 W. E. Cooper,45 M. Corcoran,75 F. Couderc,15 M.-C. Cousinou,12 A. Croc,15 D. Cutts,72 A. Das,42 G. Davies,40 S. J. de Jong,30,31 E. De La Cruz-Burelo,29 F. Déliot,15 R. Demina,66 D. Denisov,45 S. P. Denisov,35 S. Desai,45 C. Deterre,15 K. DeVaughan,61 H. T. Diehl,45 M. Diesburg,45 P. F. Ding,41 A. Dominguez,61 A. Dubey,25 L. V. Dudko,34 D. Duggan,62 A. Duperrin,12 S. Dutt,24 A. Dyshkant,47 M. Eads,61 D. Edmunds,59 J. Ellison,43 V.D. Elvira,45 Y. Enari,14 H. Evans,49 A. Evdokimov,68 V.N. Evdokimov,35 G. Facini,57 L. Feng,47 T. Ferbel,66 F. Fiedler,21 F. Filthaut,30,31 W. Fisher,59 H. E. Fisk,45 M. Fortner,47 H. Fox,39 S. Fuess,45 A. Garcia-Bellido,66 J. A. Garcı́a-González,29 G.A. Garcı́a-Guerra,29,§ V. Gavrilov,33 P. Gay,10 W. Geng,12,59 D. Gerbaudo,63 C. E. Gerber,46 Y. Gershtein,62 G. Ginther,45,66 G. Golovanov,32 A. Goussiou,77 P. D. Grannis,67 S. Greder,16 H. Greenlee,45 G. Grenier,17 Ph. Gris,10 J.-F. Grivaz,13 A. Grohsjean,15,k S. Grünendahl,45 M.W. Grünewald,27 T. Guillemin,13 G. Gutierrez,45 P. Gutierrez,70 A. Haas,65,{ S. Hagopian,44 J. Haley,57 L. Han,4 K. Harder,41 A. Harel,66 J.M. Hauptman,52 J. Hays,40 T. Head,41 T. Hebbeker,18 D. Hedin,47 H. Hegab,71 A. P. Heinson,43 U. Heintz,72 C. Hensel,20 I. Heredia-De La Cruz,29 K. Herner,58 G. Hesketh,41,** M.D. Hildreth,51 R. Hirosky,76 T. Hoang,44 J. D. Hobbs,67 B. Hoeneisen,9 M. Hohlfeld,21 I. Howley,73 Z. Hubacek,7,15 V. Hynek,7 I. Iashvili,64 Y. Ilchenko,74 R. Illingworth,45 A. S. Ito,45 S. Jabeen,72 M. Jaffré,13 A. Jayasinghe,70 R. Jesik,40 K. Johns,42 E. Johnson,59 M. Johnson,45 A. Jonckheere,45 P. Jonsson,40 J. Joshi,43 A.W. Jung,45 A. Juste,37 K. Kaadze,54 E. Kajfasz,12 D. Karmanov,34 P. A. Kasper,45 I. Katsanos,61 R. Kehoe,74 S. Kermiche,12 N. Khalatyan,45 A. Khanov,71 A. Kharchilava,64 Y. N. Kharzheev,32 I. Kiselevich,33 J.M. Kohli,24 V. A. Kostelecký,49 A.V. Kozelov,35 J. Kraus,60 S. Kulikov,35 A. Kumar,64 A. Kupco,8 T. Kurča,17 V. A. Kuzmin,34 S. Lammers,49 G. Landsberg,72 P. Lebrun,17 H. S. Lee,28 S.W. Lee,52 W.M. Lee,45 J. Lellouch,14 H. Li,11 L. Li,43 Q. Z. Li,45 J. K. Lim,28 D. Lincoln,45 J. Linnemann,59 V.V. Lipaev,35 R. Lipton,45 H. Liu,74 Y. Liu,4 A. Lobodenko,36 M. Lokajicek,8 R. Lopes de Sa,67 H. J. Lubatti,77 R. Luna-Garcia,29,†† A. L. Lyon,45 A.K. A. Maciel,1 R. Madar,15 R. Magaña-Villalba,29 S. Malik,61 V. L. Malyshev,32 Y. Maravin,54 J. Martı́nez-Ortega,29 R. McCarthy,67 C. L. McGivern,53 M.M. Meijer,30,31 A. Melnitchouk,60 D. Menezes,47 P. G. Mercadante,3 M. Merkin,34 A. Meyer,18 J. Meyer,20 F. Miconi,16 N.K. Mondal,26 M. Mulhearn,76 E. Nagy,12 M. Naimuddin,25 M. Narain,72 R. Nayyar,42 H.A. Neal,58 J. P. Negret,5 P. Neustroev,36 T. Nunnemann,22 G. Obrant,36,* J. Orduna,75 N. Osman,12 J. Osta,51 M. Padilla,43 A. Pal,73 N. Parashar,50 V. Parihar,72 S. K. Park,28 R. Partridge,72,{ N. Parua,49 A. Patwa,68 B. Penning,45 M. Perfilov,34 Y. Peters,41 K. Petridis,41 G. Petrillo,66 P. Pétroff,13 M.-A. Pleier,68 P. L.M. Podesta-Lerma,29,‡‡ V.M. Podstavkov,45 A.V. Popov,35 M. Prewitt,75 D. Price,49 N. Prokopenko,35 J. Qian,58 A. Quadt,20 B. Quinn,60 M. S. Rangel,1 K. Ranjan,25 P. N. Ratoff,39 I. Razumov,35 P. Renkel,74 I. Ripp-Baudot,16 F. Rizatdinova,71 M. Rominsky,45 A. Ross,39 C. Royon,15 P. Rubinov,45 R. Ruchti,51 G. Sajot,11 P. Salcido,47 A. Sánchez-Hernández,29 M. P. Sanders,22 B. Sanghi,45 A. S. Santos,1,§§ G. Savage,45 L. Sawyer,55 T. Scanlon,40 R. D. Schamberger,67 Y. Scheglov,36 H. Schellman,48 S. Schlobohm,77 C. Schwanenberger,41 R. Schwienhorst,59 J. Sekaric,53 H. Severini,70 E. Shabalina,20 V. Shary,15 S. Shaw,59 A.A. Shchukin,35 R. K. Shivpuri,25 V. Simak,7 P. Skubic,70 P. Slattery,66 D. Smirnov,51 K. J. Smith,64 G. R. Snow,61 J. Snow,69 S. Snyder,68 S. Söldner-Rembold,41 L. Sonnenschein,18 K. Soustruznik,6 J. Stark,11 D.A. Stoyanova,35 M. Strauss,70 L. Stutte,45 L. Suter,41 P. Svoisky,70 M. Takahashi,41 M. Titov,15 V. V. Tokmenin,32 Y.-T. Tsai,66 K. Tschann-Grimm,67 D. Tsybychev,67 B. Tuchming,15 C. Tully,63 L. Uvarov,36 S. Uvarov,36 S. Uzunyan,47 R. Van Kooten,49 W.M. van Leeuwen,30 N. Varelas,46 E.W. Varnes,42 I. A. Vasilyev,35 P. Verdier,17 A.Y. Verkheev,32 L. S. Vertogradov,32 M. Verzocchi,45 M. Vesterinen,41 D. Vilanova,15 P. Vokac,7 H. D. Wahl,44 M.H. L. S. Wang,45 J. Warchol,51 G. Watts,77 M. Wayne,51 J. Weichert,21 L. Welty-Rieger,48 A. White,73 D. Whittington,49 D. Wicke,23 M.R. J. Williams,39 G.W. Wilson,53 M. Wobisch,55 D. R. Wood,57 PRL 108, 261603 (2012) P HY S I CA L R EV I EW LE T T E R S week ending 29 JUNE 2012 0031-9007=12=108(26)=261603(7) 261603-1 � 2012 American Physical Society T. R. Wyatt,41 Y. Xie,45 R. Yamada,45 W.-C. Yang,41 T. Yasuda,45 Y. A. Yatsunenko,32 W. Ye,67 Z. Ye,45 H. Yin,45 K. Yip,68 S.W. Youn,45 J. Zennamo,64 T. Zhao,77 T.G. Zhao,41 B. Zhou,58 J. Zhu,58 M. Zielinski,66 D. Zieminska,49 and L. Zivkovic72 (The D0 Collaboration) 1LAFEX, Centro Brasileiro de Pesquisas Fı́sicas, Rio de Janeiro, Brazil 2Universidade do Estado do Rio de Janeiro, Rio de Janeiro, Brazil 3Universidade Federal do ABC, Santo André, Brazil 4University of Science and Technology of China, Hefei, People’s Republic of China 5Universidad de los Andes, Bogotá, Colombia 6Faculty of Mathematics and Physics, Center for Particle Physics, Prague, Charles University, Czech Republic 7Czech Technical University in Prague, Prague, Czech Republic 8Center for Particle Physics, Institute of Physics, Academy of Sciences of the Czech Republic, Prague, Czech Republic 9Universidad San Francisco de Quito, Quito, Ecuador 10LPC, Université Blaise Pascal, CNRS/IN2P3, Clermont, France 11LPSC, Université Joseph Fourier Grenoble 1, CNRS/IN2P3, Institut National Polytechnique de Grenoble, Grenoble, France 12CPPM, Aix-Marseille Université, CNRS/IN2P3, Marseille, France 13LAL, Université Paris-Sud, CNRS/IN2P3, Orsay, France 14LPNHE, Universités Paris VI and VII, CNRS/IN2P3, Paris, France 15CEA, Irfu, SPP, Saclay, France 16IPHC, Université de Strasbourg, CNRS/IN2P3, Strasbourg, France 17IPNL, Université Lyon 1, CNRS/IN2P3, Villeurbanne, France and Université de Lyon, Lyon, France 18III. Physikalisches Institut A, RWTH Aachen University, Aachen, Germany 19Physikalisches Institut, Universität Freiburg, Freiburg, Germany 20II. Physikalisches Institut, Georg-August-Universität Göttingen, Göttingen, Germany 21Institut für Physik, Universität Mainz, Mainz, Germany 22Ludwig-Maximilians-Universität München, München, Germany 23Fachbereich Physik, Bergische Universität Wuppertal, Wuppertal, Germany 24Panjab University, Chandigarh, India 25Delhi University, Delhi, India 26Tata Institute of Fundamental Research, Mumbai, India 27University College Dublin, Dublin, Ireland 28Korea Detector Laboratory, Korea University, Seoul, Korea 29CINVESTAV, Mexico City, Mexico 30Nikhef, Science Park, Amsterdam, The Netherlands 31Radboud University Nijmegen, Nijmegen, The Netherlands 32Joint Institute for Nuclear Research, Dubna, Russia 33Institute for Theoretical and Experimental Physics, Moscow, Russia 34Moscow State University, Moscow, Russia 35Institute for High Energy Physics, Protvino, Russia 36Petersburg Nuclear Physics Institute, St. Petersburg, Russia 37Institució Catalana de Recerca i Estudis Avançats (ICREA) and Institut de Fı́sica d’Altes Energies (IFAE), Barcelona, Spain 38Uppsala University, Uppsala, Sweden 39Lancaster University, Lancaster LA1 4YB, United Kingdom 40Imperial College London, London SW7 2AZ, United Kingdom 41The University of Manchester, Manchester M13 9PL, United Kingdom 42University of Arizona, Tucson, Arizona 85721, USA 43University of California Riverside, Riverside, California 92521, USA 44Florida State University, Tallahassee, Florida 32306, USA 45Fermi National Accelerator Laboratory, Batavia, Illinois 60510, USA 46University of Illinois at Chicago, Chicago, Illinois 60607, USA 47Northern Illinois University, DeKalb, Illinois 60115, USA 48Northwestern University, Evanston, Illinois 60208, USA 49Indiana University, Bloomington, Indiana 47405, USA 50Purdue University Calumet, Hammond, Indiana 46323, USA 51University of Notre Dame, Notre Dame, Indiana 46556, USA 52Iowa State University, Ames, Iowa 50011, USA 53University of Kansas, Lawrence, Kansas 66045, USA PRL 108, 261603 (2012) P HY S I CA L R EV I EW LE T T E R S week ending 29 JUNE 2012 261603-2 54Kansas State University, Manhattan, Kansas 66506, USA 55Louisiana Tech University, Ruston, Louisiana 71272, USA 56Boston University, Boston, Massachusetts 02215, USA 57Northeastern University, Boston, Massachusetts 02115, USA 58University of Michigan, Ann Arbor, Michigan 48109, USA 59Michigan State University, East Lansing, Michigan 48824, USA 60University of Mississippi, University, Mississippi 38677, USA 61University of Nebraska, Lincoln, Nebraska 68588, USA 62Rutgers University, Piscataway, New Jersey 08855, USA 63Princeton University, Princeton, New Jersey 08544, USA 64State University of New York, Buffalo, New York 14260, USA 65Columbia University, New York, New York 10027, USA 66University of Rochester, Rochester, New York 14627, USA 67State University of New York, Stony Brook, New York 11794, USA 68Brookhaven National Laboratory, Upton, New York 11973, USA 69Langston University, Langston, Oklahoma 73050, USA 70University of Oklahoma, Norman, Oklahoma 73019, USA 71Oklahoma State University, Stillwater, Oklahoma 74078, USA 72Brown University, Providence, Rhode Island 02912, USA 73University of Texas, Arlington, Texas 76019, USA 74Southern Methodist University, Dallas, Texas 75275, USA 75Rice University, Houston, Texas 77005, USA 76University of Virginia, Charlottesville, Virginia 22901, USA 77University of Washington, Seattle, Washington 98195, USA (Received 29 March 2012; published 27 June 2012) Using data collected with the D0 detector at the Fermilab Tevatron Collider, corresponding to 5:3 fb�1 of integrated luminosity, we search for violation of Lorentz invariance by examining the t�t production cross section in leptonþ jets final states. We quantify this violation using the standard-model extension framework, which predicts a dependence of the t�t production cross section on sidereal time as the orientation of the detector changes with the rotation of the Earth. Within this framework, we measure components of the matrices ðcQÞ��33 and ðcUÞ��33 containing coefficients used to parametrize violation of Lorentz invariance in the top quark sector. Within uncertainties, these coefficients are found to be consistent with zero. DOI: 10.1103/PhysRevLett.108.261603 PACS numbers: 11.30.Cp, 13.85.Qk, 14.65.Ha We investigate the possibility of Lorentz-invariance violation (LIV) in the top quark (t) sector, using data collected with the D0 detector at the Fermilab Tevatron p �p Collider corresponding to 5:3 fb�1 of integrated luminosity collected between August 2002 and June 2009. We examine events in which a t�t pair is produced and decays into a final state, including two light quarks ( �q, q0), two b quarks (b, �b), and a lepton-neutrino pair (‘, �‘) via the mode t�t ! WþbW� �b ! ‘�‘b �qq 0 �b, where ‘ ¼ e;�. The standard-model extension (SME) frame- work [1] provides an effective field theoretical treatment for the violation of Lorentz and CPT symmetry in particle interactions by introducing Lorentz-violating terms to the Lagrangian density of the standard model (SM). As yet, there are no quantitative limits on viola- tions of CPT or Lorentz invariance in the top quark sector [2]. This parameter space is accessible only at high-energy particle colliders. Because top quarks decay before hadronizing, this Letter also offers the possibility of extending such an investigation to what are essentially free quarks. Strong limits have been set on the magnitude of LIV in gravitational and electromagnetic interactions, as well as in many particle sectors. Most constraints in the particle sectors are for matter involving quarks of the first genera- tion. There are also sensitive limits on SME coefficients for the second generation, but only a few for the third genera- tion [2]. The latter include limits for the b quark through B-meson oscillations [3] and for � neutrinos from neutrino oscillations [4]. There is also a constraint for � leptons deduced from theoretical grounds using astrophysical ob- servations [5]. Constraints on LIV have also been predicted for the SM Higgs sector, as derived from radiative correc- tions [6]. Many of the limits on LIV coefficients in the quark, lepton, and gauge boson sectors are & 10�5. However, no constraints have yet been placed on LIV in the top quark sector. Because the SME represents a general phenomenological formalism, the LIV terms of the SME are not constrained to couple with the same strength to all particle species. We therefore consider separately only those SME terms that affect the top quark fields in t�t events. PRL 108, 261603 (2012) P HY S I CA L R EV I EW LE T T E R S week ending 29 JUNE 2012 261603-3 http://dx.doi.org/10.1103/PhysRevLett.108.261603 While it has been shown that CPT violation implies violation of Lorentz invariance [7], the contributions from CPT-violating terms in the SME to the matrix ele- ment for t�t production and decay are suppressed. However, contributions from other Lorentz-violating terms can be significant [8]. At leading order in LIV coefficients, the matrix element describing the production and decay of a t�t pair involves coefficients of the form c��, where � and � refer to space-time indices. Although at leading order CPT-odd SME terms describing LIV in the top quark sector are not observable in t�t production or decay, this analysis is sensitive to several components of the CPT-even ðcQÞ��AB and ðcUÞ��AB terms, where A; B ¼ 3; 3 refer to the third quark generation. The ðcQÞ��33 are the SME coefficients coupling to the left-handed compo- nents of the third generation quark fields, and ðcUÞ��33 are the SME coefficients coupling to the right-handed singlet top quark field. For brevity, we drop the generation sub- scripts since we are restricting the analysis to the terms that couple to the top quark fields. To compare our results with SME studies in other particle sectors [2], we also examine the linear combinations c��¼ðcQÞ��þðcUÞ��; d��¼ðcQÞ���ðcUÞ��: (1) The matrix element for leading-order t�t production and decay, including leading-order contributions from SME terms, can be written as [8] jMj2SME ¼ PF �Fþ ð�PÞF �Fþ Pð�FÞ �Fþ PFð� �FÞ: (2) The P terms are functions of the parton momenta at the t�t production vertex, while the F terms involve parton mo- menta at the decay vertices. The PF �F term corresponds to the usual SM component, while the � terms reflect the dependence on SME coefficients. This expression summa- rizes how the SME modifies the matrix element for t�t production and decay at leading order. The � terms contain contractions of c�� coefficients with tensors that are functions of the four-momenta of the particles in t�t production and decay. Because of the V-A structure of the weak current, the right-handed coef- ficients, ðcUÞ��, couple only to the production (�P) terms, while the left-handed coefficients, ðcQÞ��, couple to both production and decay (�F) terms. The matrices of c�� coefficients are symmetric and traceless. Within the SME, these coefficients are defined by convention in the canonical Sun-centered reference frame [2]. The kinematic component of the � terms of Eq. (2) can be evaluated in any coordinate system. A convenient ref- erence frame is that of a coordinate system fixed to the measuring apparatus, and we therefore choose to evaluate such contractions in the D0 coordinate system. In this system, the momenta entering the calculation of Eq. (2) are just the momenta of the particles measured in the detector, and, to calculate the matrix element, the coefficients ðcUÞ�� and ðcQÞ�� must therefore be trans- formed from the Sun’s reference system to the D0 coor- dinate system. Since the Earth is rotating about its axis, the trans- formation of the coefficients ðcUÞ�� and ðcQÞ�� from the Sun-centered frame to the laboratory frame introduces a time dependence. The relevant time scale is the sidereal day, which has a period of 23 hr 56 min 4.1 s (86 164.1 s). Effects due to the motion of the Earth around the Sun correspond to corrections of the order of 10�4 to these sidereal variations and are therefore neglected. If any of the coefficients ðcUÞ�� or ðcQÞ�� are nonzero in the Sun-centered frame, they can be detected through a periodic oscillation in the number of t�t events observed in the Earth-based detector as a func- tion of sidereal time. The data used for this analysis correspond to 5:3 fb�1 of integrated luminosity collected with the D0 detector. The D0 detector [9] consists of several subdetectors designed for identification and reconstruction of the products of p �p collisions. A silicon microstrip tracker and central fiber tracker surround the interaction region for pseudorapidities j�j< 3 and j�j< 2:5, respectively {where � ¼ � ln½tanð�=2Þ� is measured relative to the center of the detector and � is the polar angle with respect to the proton beam direction}. These elements of the central tracking system are located within a 2 T superconducting solenoidal magnet, providing measurements for reconstructing event vertices and paths of charged particles. Particle energies are measured using a liquid argon and uranium calorimeter. Outside of the calorimetry, trajectories of muons are mea- sured using three layers of tracking detectors and scintil- lation trigger counters, with 1.8 T iron toroidal magnets between the first two layers. Plastic scintillator arrays in front of the end-calorimeter cryostats provide measure- ments of luminosity. We employ the same event selection as described in greater detail in Ref. [10]. Briefly, events are collected using a suite of triggers selecting events with a single lepton (e or �) or a single lepton plus a jet. Candidate t�t events in the leptonþ jets channels are then selected by requiring the presence of one isolated electron (or muon) candidate with transverse momentum pT > 20 GeV and pseudorapidity j�j< 1:1 (2.0) and an imbalance in trans- verse energy of 6ET > 20 GeV (25 GeV). Events are di- vided into bins of jet multiplicity, and all jets are required to be reconstructed with pT > 20 GeV and j�j< 2:5, with a leading jet of pT > 40 GeV. One of the jets is required to be tagged as a b-jet candidate through a neural-network- based algorithm [11]. The time of production of each t�t event is recorded with the event data, with an average accuracy of approximately �30 s. To follow the conven- tions utilized in other SME studies [2], we shift the origin of the time coordinate to correspond to the vernal equinox of the year 2000. PRL 108, 261603 (2012) P HY S I CA L R EV I EW LE T T E R S week ending 29 JUNE 2012 261603-4 The SME predicts time-dependent effects on the t�t cross section of the form �ðtÞ � �av½1þ fSMEðtÞ�; (3) where�av is the observed (time-averaged) cross section for t�t ! WþbW� �b ! ‘�‘b �qq 0 �b, in ‘þ jets final states. To arrive at Eq. (3), we compare the contribution from the SME terms in Eq. (2) to the SM expectation by considering the ratio of jMj2SME to the SM component PF �F. The SME contributions in this ratio are collected into the function fSMEðtÞ ¼ ½ðcQÞ�� þ ðcUÞ���R� � ðtÞR� ðtÞA� P þ ðcQÞ��R � � ðtÞR� ðtÞA� F : (4) Equation (4) is a product of the matrices of time- independent coefficients ðcQÞ�� and ðcUÞ��, four-by-four matrices of terms that depend on the event production (A� P ) and decay (A� F ) kinematics in the D0 frame, and a rotation matrix R� � ðtÞ that transforms A� P and A� F from the D0 frame to the Sun-centered frame. The A� P and A� F matrices are evaluated using t�t Monte Carlo events generated with PYTHIA [12]. Events that pass detector acceptance, trigger, event reconstruction, and analysis selections (modeled by a full simulation of the D0 detector) are corrected according to the SME expecta- tion of Eq. (2). The SME contribution to the cross section has the gen- eral form fSMEðtÞ ¼ C��R � � ðtÞR� ðtÞA� for the four model assumptions summarized in Table I. For each model, C�� represents the constant coefficients we wish to determine and A� refers to the appropriate linear combination of A� P and A� F . For each model, we estimate one possible component of C�� at a time. We impose the requirements that each tensor C�� is symmetric and traceless, choosing CXX ¼ �CYY to satisfy the latter condition. We adopt the index ordering conventions �; � ¼ fT; X; Y; Zg to refer to coordinates in the Sun-centered frame and �; ¼ ft; x; y; zg for coordi- nates in the D0 frame. Evaluating Eq. (4) for the different assumptions of Table I yields the following results: (i) Coefficients CTT and CZZ contribute only to the total cross section, and we do not attempt to extract these coefficients. (ii) Coefficients CTX, CTY , and CTZ combine with the small off-diagonal elements of matrices A� P and A� F , for which we expect poor sensitivity. (iii) Coefficients CXZ and CYZ couple to expressions that depend on sidereal time (differing by a phase of =2). (iv) Coefficients CXX and CXY couple to time-dependent expressions with twice the sidereal frequency, and the two terms differ by a phase of =4. Table II collects the resulting forms of the function fSMEðtÞ for different assumptions. We refer to the ‘‘sidereal phase’’ !st as �, where !s is the inverse of the sidereal day. The b terms in these expressions depend on the colatitude of the detector, the orientation of the proton beam at the detector relative to geographic north, and the XX and ZZ elements of the combination of A� P and A� F that are appropriate to the particular assumption of the model. Assuming that any LIVoriginates from just the top quark sector, we expect the background rate (principally W þ jets events) to be proportional only to the luminosity. To search for a signal varying with sidereal time, we sum the contributions to each of 12 Ni bins (corresponding to two sidereal hours each) for all data: Ni � Ntot Li Lint ½1þ fSfSMEð�iÞ�; (5) whereNtot is the total number of signal (t�t) and background (non-t�t) events corresponding to the total integrated lumi- nosity Lint, Li is the integrated luminosity over the appro- priate bin of sidereal phase �i, and fS is the average fraction of signal events in the data. We extract fS from the data that were used previously to determine the t�t cross section in ‘þ jets events [10]. The t�t cross section is measured in bins of jet multiplicity for the eþ jets and �þ jets channels. The subset of events with at least four reconstructed jets that pass selection require- ments contains a high fraction of t�t events, providing the best sensitivity to any time dependence in the t�t event rate. We find fSðeþ>3 jetsÞ ¼ 0:78� 0:12 and fSð�þ>3 jetsÞ ¼ 0:76� 0:11. Because of this differ- ence, we treat the electron and muon channels separately. To simplify fitting fSMEð�Þ to the data, we define a variable R for each bin: Ri � 1 fS � Ni=Ntot Li=Lint � 1 � : (6) Equation (6) is the luminosity-corrected sidereally binned relative t�t event rate, which can be compared directly to TABLE I. fSMEðtÞ for different SME assumptions. Assumption fSMEðtÞ ðcUÞ�� ¼ 0 ðcQÞ��R � � ðtÞR� ðtÞðA� P þ A� F Þ ðcQÞ�� ¼ 0 ðcUÞ��R � � ðtÞR� ðtÞðA� P Þ c�� ¼ 0 d��R � � ðtÞR� ðtÞ 12A� F d�� ¼ 0 c��R � � ðtÞR� ðtÞðA� P þ 1 2A � F Þ TABLE II. Forms for fSMEð�Þ used to extract SME coefficients. Condition fSMEð�Þ CXX ¼ �CYY 2CXXðb1�b2 2 cos2�þ b3 sin2�Þ CXY ¼ CYX 2CXYðb1�b2 2 sin2�� b3 cos2�Þ CXZ ¼ CZX 2CXZðb4 cos�þ b5 sin�Þ CYZ ¼ CZY 2CYZðb4 sin�� b5 cos�Þ PRL 108, 261603 (2012) P HY S I CA L R EV I EW LE T T E R S week ending 29 JUNE 2012 261603-5 fSMEð�Þ. In the absence of any significant sidereal time dependence, all the Ri values should be consistent with zero, while a sidereal time dependence would produce a sinusoidal variation in this rate. The amplitude for any sinusoidal dependence is given by the product of an SME coefficient and a mixture of contributions from the rotation matrix and an appropriate combination of elements from A� P and A� F . This latter mixture also fixes the phase of the sinusoidal function in a fit to the data. The resulting distributions for R as a function of sidereal phase are shown in Fig. 1, separately for the electron and muon channels. The forms of fSMEð�Þ are fitted to these two distributions to estimate the values of the SME coef- ficients for the assumptions summarized in Tables I and II. We apply a small correction of 1:2%–4:7% to each ex- tracted value to account for biases introduced by the finite bin size. While the dominant contribution to the uncertainty on the SME coefficients results from the limited size of our t�t data sample, the estimated fraction of t�t events in the data contributes an additional uncertainty. We treat this as a systematic uncertainty. The background from single top quark events can, in principle, exhibit SME effects. However, their relative contribution to the t�t sample is negligible (� 1%). The orientation and location of the detector, as well as the origin chosen for the time of events, also carry negligible uncertainties. Finally, any uncertain- ties in the values of the elements of A� P and A� F can potentially contribute a systematic uncertainty in this analysis, in ways similar to those discussed in the analysis of the t�t cross section, as summarized below. The leading sources of systematic uncertainty in the kinematics of t�t events arise from (i) the jet energy scale, (ii) jet energy resolution, and (iii) jet identification. These can affect the distributions of momenta reconstructed in the detector, but, as the elements of A� P and A� F reflect only average values of the components of the momenta over the detector acceptance, such averages are not very sensitive to small changes in kinematic parameters. The relative un- certainty of the contributing elements is negligible com- pared to the statistical uncertainty of the data and the systematic uncertainties on signal fractions fS. A periodic time dependence could potentially be intro- duced to the event rate through changes in event selection efficiency. Various environmental effects ranging from day/night temperature cycling to accelerator conditions could contribute to changes in the efficiency of various detector elements. We check this possibility by examining the luminosity-corrected sidereally binned relative event rates (R distributions) for the leptonþ n jets channels, where n ¼ 2; 3. These bins of jet multiplicity contain relatively small contributions from t�t events, with fSð‘þ 2 jetsÞ � 12% and fSð‘þ 3 jetsÞ � 45%, and consist mostly of W þ jets events in which we expect no sidereal effects from LIV. Such events are topologically similar to the signal t�t events and therefore provide a handle on detector efficiencies that could affect our measurement. We extract the amplitudes for any time-dependent oscilla- tions, corresponding to the parametrizations used for the coefficients in Tables III, IV, and V, in each of the four cross-check channels (‘þ n jets, where ‘ ¼ e;� and n ¼ 2; 3). For each assumption, those amplitudes are less than 10% and the ensemble of fits is consistent with no time dependence at levels of probability in the range 6%–38%. We therefore conclude that these cross-checks give no indication of a sidereal time-dependent efficiency. Finally, it should be noted that any residual nonsidereal time dependence is suppressed greatly by folding the data into 12 bins of sidereal phase, as the magnitude of any residual contribution following this folding depends in- versely on the difference in the period of the time- dependent efficiency and the sidereal period. Most prob- lematic would be an unexpected time-dependent efficiency with a period close to that of a sidereal day. The worst realistic case would be a contribution to detector efficiency that has a period of 24 solar hours. However, because the πSidereal Phase / 2 0 0.25 0.5 0.75 1 R -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 -1DØ, 5.3 fb channele / tt (a) πSidereal Phase / 2 0 0.25 0.5 0.75 1 R -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 -1DØ, 5.3 fb channelµ / tt (b) FIG. 1. The dependence of R, as defined in Eq. (6), on the sidereal phase for (a) eþ>3 jets t�t candidates and (b) �þ>3 jets t�t candidates. TABLE III. Limits on SME coefficients at the 95% C.L., assuming ðcUÞ�� � 0. Coefficient Value� Stat� Sys 95% C.L. Interval ðcQÞXX33 �0:12� 0:11� 0:02 [� 0:34, þ0:11] ðcQÞYY33 0:12� 0:11� 0:02 [� 0:11, þ0:34] ðcQÞXY33 �0:04� 0:11� 0:01 [� 0:26, þ0:18] ðcQÞXZ33 0:15� 0:08� 0:02 [� 0:01, þ0:31] ðcQÞYZ33 �0:03� 0:08� 0:01 [� 0:19, þ0:12] TABLE IV. Limits on SME coefficients at the 95% C.L., as- suming ðcQÞ�� � 0. Coefficient Value� Stat� Sys 95% C.L. Interval ðcUÞXX33 0:10� 0:09� 0:02 [� 0:08, þ0:27] ðcUÞYY33 �0:10� 0:09� 0:02 [� 0:27, þ0:08] ðcUÞXY33 0:04� 0:09� 0:01 [� 0:14, þ0:22] ðcUÞXZ33 �0:14� 0:07� 0:02 [� 0:28, þ0:01] ðcUÞYZ33 0:01� 0:07�<0:01 [� 0:13, þ0:14] PRL 108, 261603 (2012) P HY S I CA L R EV I EW LE T T E R S week ending 29 JUNE 2012 261603-6 data taking spans approximately seven years, any contri- butions from such an effect would be suppressed by about a factor of 10. To affect our conclusions, we would have had to experience a highly unlikely periodic dependence of the efficiency of approximately 75% over 24 hours. No peri- odic effects of this magnitude have ever been observed in the detection efficiencies for objects considered in this analysis. Because the SME contribution to the matrix element is independent of lepton flavor, we perform a simultaneous fit to both the eþ>3 jets and �þ>3 jets data to obtain the final results. The extracted SME coefficients are all con- sistent with no time dependence, and we therefore find no evidence for violation of Lorentz invariance in the t�t system. We define the observed limits (95% C.L. intervals) for each SME coefficient as the extracted value �2 standard deviations. Because the magnitudes of the 95% confidence bounds on elements of the linear combination c�� ¼ ðcQÞ�� þ ðcUÞ�� for the assumption of d�� ¼ 0 are larger than 1, we cannot place meaningful limits on these combi- nations of SME coefficients in this analysis. The remaining limits are presented in Tables III, IV, and V. In the SME, different particles can have distinct Lorentz-violating properties, so it is of interest to test all species. Most constraints on LIV are for particles of the first and second generations, with a few limits on SME coefficients for the third generation. The only sector for which no constraints on Lorentz violation exist to date is the top quark [2]. The limits on the ðcQÞ��33 and ðcUÞ��33 coefficients determined in this Letter represent the first constraints on LIV in the top quark sector and the first such constraints on any free quark. We thank the staffs at Fermilab and collaborating insti- tutions and acknowledge support from the DOE and NSF (USA); CEA and CNRS/IN2P3 (France); MON, Rosatom, and RFBR (Russia); CNPq, FAPERJ, FAPESP, and FUNDUNESP (Brazil); DAE and DST (India); Colciencias (Colombia); CONACyT (Mexico); NRF (Korea); FOM (The Netherlands); STFC and the Royal Society (United Kingdom); MSMT and GACR (Czech Republic); BMBF and DFG (Germany); SFI (Ireland); The Swedish Research Council (Sweden); and CAS and CNSF (China). We also acknowledge support from the Indiana University Center for Spacetime Symmetries (IUCSS). *Deceased. †Visitor from Augustana College, Sioux Falls, SD 57197, USA. ‡Visitor from The University of Liverpool, Liverpool, UK. §Visitor from UPIITA-IPN, Mexico City, Mexico. kVisitor from DESY, Hamburg, Germany. {Visitor from SLAC, Menlo Park, CA 94025, USA. **Visitor from University College London, London, UK. ††Visitor from Centro de Investigacion en Computacion- IPN, Mexico City, Mexico. ‡‡Visitor from ECFM, Universidad Autonoma de Sinaloa, Culiacán, Mexico. §§Visitor from Universidade Estadual Paulista, São Paulo, Brazil. [1] D. Colladay and V.A. Kostelecký, Phys. Rev. D 58, 116002 (1998); V.A. Kostelecký, ibid. 69, 105009 (2004). [2] V. A. Kostelecký and N. Russell, Rev. Mod. Phys. 83, 11 (2011). [3] B. Aubert et al. (BABAR Collaboration), Phys. Rev. Lett. 100, 131802 (2008); V. A. Kostelecký and R. J. Van Kooten, Phys. Rev. D 82, 101702(R) (2010). [4] P. Adamson et al. (MINOS Collaboration), Phys. Rev. Lett. 101, 151601 (2008); 105, 151601 (2010); Phys. Rev. D 85, 031101 (2012); R. Abbasi et al. (IceCube Collaboration), ibid. 82, 112003 (2010). [5] B. Altschul, Astropart. Phys. 28, 380 (2007). [6] D. L. Anderson, M. Sher, and I. Turan, Phys. Rev. D 70, 016001 (2004). [7] O.W. Greenberg, Phys. Rev. Lett. 89, 231602 (2002). [8] M. Berger, in Proceedings of the Fifth Meeting on CPT and Lorentz Symmetry, edited by V.A. Kostelecký (World Scientific, Hackensack, NJ, 2010). [9] V.M. Abazov et al. (D0 Collaboration), Nucl. Instrum. Methods Phys. Res., Sect. A 565, 463 (2006); M. Abolins et al., ibid. 584, 75 (2008); R. Angstadt et al. (D0 Collaboration), ibid. 622, 298 (2010). [10] V.M. Abazov et al. (D0 Collaboration), Phys. Rev. D 84, 012008 (2011). [11] V.M. Abazov et al. (D0 Collaboration), Nucl. Instrum. Methods Phys. Res., Sect. A 620, 490 (2010). [12] T. Sjöstrand, S. Mrenna, and P. Skands, J. High Energy Phys. 05 (2006) 026. TABLE V. Limits on SME coefficients at the 95% C.L., as- suming c�� � 0. Coefficient Value� Stat� Sys 95% C.L. Interval dXX �0:11� 0:10� 0:02 [� 0:31, þ0:09] dYY 0:11� 0:10� 0:02 [� 0:09, þ0:31] dXY �0:04� 0:10� 0:01 [� 0:24, þ0:16] dXZ 0:14� 0:07� 0:02 [� 0:01, þ0:29] dYZ �0:02� 0:07�<0:01 [� 0:16, þ0:13] PRL 108, 261603 (2012) P HY S I CA L R EV I EW LE T T E R S week ending 29 JUNE 2012 261603-7 http://dx.doi.org/10.1103/PhysRevD.58.116002 http://dx.doi.org/10.1103/PhysRevD.58.116002 http://dx.doi.org/10.1103/PhysRevD.69.105009 http://dx.doi.org/10.1103/RevModPhys.83.11 http://dx.doi.org/10.1103/RevModPhys.83.11 http://dx.doi.org/10.1103/PhysRevLett.100.131802 http://dx.doi.org/10.1103/PhysRevLett.100.131802 http://dx.doi.org/10.1103/PhysRevD.82.101702 http://dx.doi.org/10.1103/PhysRevLett.101.151601 http://dx.doi.org/10.1103/PhysRevLett.101.151601 http://dx.doi.org/10.1103/PhysRevLett.105.151601 http://dx.doi.org/10.1103/PhysRevD.85.031101 http://dx.doi.org/10.1103/PhysRevD.85.031101 http://dx.doi.org/10.1103/PhysRevD.82.112003 http://dx.doi.org/10.1016/j.astropartphys.2007.08.003 http://dx.doi.org/10.1103/PhysRevD.70.016001 http://dx.doi.org/10.1103/PhysRevD.70.016001 http://dx.doi.org/10.1103/PhysRevLett.89.231602 http://dx.doi.org/10.1016/j.nima.2006.05.248 http://dx.doi.org/10.1016/j.nima.2006.05.248 http://dx.doi.org/10.1016/j.nima.2007.10.014 http://dx.doi.org/10.1016/j.nima.2010.04.148 http://dx.doi.org/10.1103/PhysRevD.84.012008 http://dx.doi.org/10.1103/PhysRevD.84.012008 http://dx.doi.org/10.1016/j.nima.2010.03.118 http://dx.doi.org/10.1016/j.nima.2010.03.118 http://dx.doi.org/10.1088/1126-6708/2006/05/026 http://dx.doi.org/10.1088/1126-6708/2006/05/026