Combination of CDF and D0 measurements of theW boson helicity in top quark decays T. Aaltonen,12 V.M. Abazov,48 B. Abbott,113 B. S. Acharya,31 M. Adams,78 T. Adams,74 G.D. Alexeev,48 G. Alkhazov,52 A. Alton,96,a B. Álvarez González,57,b G. Alverson,92 S. Amerio,35a D. Amidei,96 A. Anastassov,76,c A. Annovi,34 J. Antos,53 M. Aoki,76 G. Apollinari,76 J. A. Appel,76 T. Arisawa,41 A. Artikov,48 J. Asaadi,120 W. Ashmanskas,76 A. Askew,74 B. Åsman,58 S. Atkins,89 O. Atramentov,103 B. Auerbach,72 K. Augsten,9 A. Aurisano,120 C. Avila,7 F. Azfar,66 F. Badaud,13 W. Badgett,76 T. Bae,43 L. Bagby,76 B. Baldin,76 D.V. Bandurin,74 S. Banerjee,31 A. Barbaro-Galtieri,68 E. Barberis,92 P. Baringer,87 V. E. Barnes,85 B. A. Barnett,90 J. Barreto,2 P. Barria,36a,36c J. F. Bartlett,76 P. Bartos,53 U. Bassler,18 M. Bauce,35b,35a V. Bazterra,78 A. Bean,87 F. Bedeschi,36a M. Begalli,2 S. Behari,90 C. Belanger-Champagne,58 L. Bellantoni,76 G. Bellettini,36b,36a J. Bellinger,126 D. Benjamin,110 A. 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Zucchelli33b,33a (CDF and D0 Collaborations) 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 4Institute of Particle Physics, McGill University, Montréal, Québec, Canada H3A 2T8; Simon Fraser University, Burnaby, British Columbia, Canada V5A 1S6; University of Toronto, Toronto, Ontario, Canada M5S 1A7; and TRIUMF, Vancouver, British Columbia, Canada V6T 2A3 5University of Science and Technology of China, Hefei, People’s Republic of China 6Institute of Physics, Academia Sinica, Taipei, Taiwan 11529, Republic of China 7Universidad de los Andes, Bogotá, Colombia 8Charles University, Faculty of Mathematics and Physics, Center for Particle Physics, Prague, Czech Republic 9Czech Technical University in Prague, Prague, Czech Republic 10Center for Particle Physics, Institute of Physics, Academy of Sciences of the Czech Republic, Prague, Czech Republic 11Universidad San Francisco de Quito, Quito, Ecuador 12Division of High Energy Physics, Department of Physics, University of Helsinki and Helsinki Institute of Physics, FIN-00014, Helsinki, Finland 13LPC, Université Blaise Pascal, CNRS/IN2P3, Clermont, France 14LPSC,Université Joseph Fourier Grenoble 1, CNRS/IN2P3, Institut National Polytechnique de Grenoble, Grenoble, France 15CPPM, Aix-Marseille Université, CNRS/IN2P3, Marseille, France 16LAL, Université Paris-Sud, CNRS/IN2P3, Orsay, France 17LPNHE, Universités Paris VI and VII, CNRS/IN2P3, Paris, France 18CEA, Irfu, SPP, Saclay, France 19IPHC, Université de Strasbourg, CNRS/IN2P3, Strasbourg, France 20IPNL, Université Lyon 1, CNRS/IN2P3, Villeurbanne, France, and Université de Lyon, Lyon, France 21III. Physikalisches Institut A, RWTH Aachen University, Aachen, Germany 22Physikalisches Institut, Universität Freiburg, Freiburg, Germany 23II. Physikalisches Institut, Georg-August-Universität Göttingen, Göttingen, Germany 24Institut für Experimentelle Kernphysik, Karlsruhe Institute of Technology, D-76131 Karlsruhe, Germany 25Institut für Physik, Universität Mainz, Mainz, Germany 26Ludwig-Maximilians-Universität München, München, Germany 27Fachbereich Physik, Bergische Universität Wuppertal, Wuppertal, Germany 28University of Athens, 157 71 Athens, Greece 29Panjab University, Chandigarh, India 30Delhi University, Delhi, India 31Tata Institute of Fundamental Research, Mumbai, India 32University College Dublin, Dublin, Ireland 33aIstituto Nazionale di Fisica Nucleare Bologna, Italy 33bUniversity of Bologna, I-40127 Bologna, Italy 34Laboratori Nazionali di Frascati, Istituto Nazionale di Fisica Nucleare, I-00044 Frascati, Italy 35aIstituto Nazionale di Fisica Nucleare, Sezione di Padova-Trento, Italy 35bUniversity of Padova, I-35131 Padova, Italy 36aIstituto Nazionale di Fisica Nucleare Pisa, I-56127 Pisa, Italy 36bUniversity of Pisa, I-56127 Pisa, Italy COMBINATION OF CDF AND D0 MEASUREMENTS OF . . . PHYSICAL REVIEW D 85, 071106(R) (2012) RAPID COMMUNICATIONS 071106-3 36cUniversity of Siena, I-56127 Pisa, Italy 36dScuola Normale Superiore, I-56127 Pisa, Italy 37aIstituto Nazionale di Fisica Nucleare, Sezione di Roma 1, I-00185 Roma, Italy 37bSapienza Università di Roma, I-00185 Roma, Italy 38aIstituto Nazionale di Fisica Nucleare Trieste/Udine, I-34100 Trieste, Italy 38bUniversity of Udine, I-33100 Udine, Italy 39Okayama University, Okayama 700-8530, Japan 40Osaka City University, Osaka 588, Japan 41Waseda University, Tokyo 169, Japan 42University of Tsukuba, Tsukuba, Ibaraki 305, Japan 43Center for High Energy Physics, Kyungpook National University, Daegu 702-701, Korea; Seoul National University, Seoul 151-742, Korea; Sungkyunkwan University, Suwon 440-746, Korea; Korea Institute of Science and Technology Information, Daejeon 305-806, Korea; Chonnam National University, Gwangju 500-757, Korea; and Chonbuk National University, Jeonju 561-756, Korea 44Korea Detector Laboratory, Korea University, Seoul, Korea 45CINVESTAV, Mexico City, Mexico 46Nikhef, Science Park, Amsterdam, the Netherlands 47Radboud University Nijmegen, Nijmegen, The Netherlands 48Joint Institute for Nuclear Research, Dubna, Russia 49Institute for Theoretical and Experimental Physics, Moscow, Russia 50Moscow State University, Moscow, Russia 51Institute for High Energy Physics, Protvino, Russia 52Petersburg Nuclear Physics Institute, St. Petersburg, Russia 53Comenius University, 842 48 Bratislava, Slovakia; Institute of Experimental Physics, 040 01 Kosice, Slovakia 54Institut de Fisica d’Altes Energies, ICREA, Universitat Autonoma de Barcelona, E-08193, Bellaterra (Barcelona), Spain 55Institució Catalana de Recerca i Estudis Avançats (ICREA), and Institut de Fı́sica d’Altes Energies (IFAE), Barcelona, Spain 56Centro de Investigaciones Energeticas Medioambientales y Tecnologicas, E-28040 Madrid, Spain 57Instituto de Fisica de Cantabria, CSIC-University of Cantabria, 39005 Santander, Spain 58Stockholm University, Stockholm, and Uppsala University, Uppsala, Sweden 59University of Geneva, CH-1211 Geneva 4, Switzerland 60Glasgow University, Glasgow G12 8QQ, United Kingdom 61Lancaster University, Lancaster LA1 4YB, United Kingdom 62University of Liverpool, Liverpool L69 7ZE, United Kingdom 63Imperial College London, London SW7 2AZ, United Kingdom 64University College London, London WC1E 6BT, United Kingdom 65The University of Manchester, Manchester M13 9PL, United Kingdom 66University of Oxford, Oxford OX1 3RH, United Kingdom 67University of Arizona, Tucson, Arizona 85721, USA 68Ernest Orlando Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA 69University of California, Davis, Davis, California 95616, USA 70University of California, Los Angeles, Los Angeles, California 90024, USA 71University of California Riverside, Riverside, California 92521, USA 72Yale University, New Haven, Connecticut 06520, USA 73University of Florida, Gainesville, Florida 32611, USA 74Florida State University, Tallahassee, Florida 32306, USA 75Argonne National Laboratory, Argonne, Illinois 60439, USA 76Fermi National Accelerator Laboratory, Batavia, Illinois 60510, USA 77Enrico Fermi Institute, University of Chicago, Chicago, Illinois 60637, USA 78University of Illinois at Chicago, Chicago, Illinois 60607, USA 79Northern Illinois University, DeKalb, Illinois 60115, USA 80Northwestern University, Evanston, Illinois 60208, USA 81University of Illinois, Urbana, Illinois 61801, USA 82Indiana University, Bloomington, Indiana 47405, USA 83Purdue University Calumet, Hammond, Indiana 46323, USA 84University of Notre Dame, Notre Dame, Indiana 46556, USA 85Purdue University, West Lafayette, Indiana 47907, USA 86Iowa State University, Ames, Iowa 50011, USA 87University of Kansas, Lawrence, Kansas 66045, USA 88Kansas State University, Manhattan, Kansas 66506, USA T. 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PHYSICAL REVIEW D 85, 071106(R) (2012) RAPID COMMUNICATIONS 071106-4 89Louisiana Tech University, Ruston, Louisiana 71272, USA 90The Johns Hopkins University, Baltimore, Maryland 21218, USA 91Boston University, Boston, Massachusetts 02215, USA 92Northeastern University, Boston, Massachusetts 02115, USA 93Harvard University, Cambridge, Massachusetts 02138, USA 94Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA 95Tufts University, Medford, Massachusetts 02155, USA 96University of Michigan, Ann Arbor, Michigan 48109, USA 97Wayne State University, Detroit, Michigan 48201, USA 98Michigan State University, East Lansing, Michigan 48824, USA 99University of Mississippi, University, Mississippi 38677, USA 100University of Nebraska, Lincoln, Nebraska 68588, USA 101University of New Mexico, Albuquerque, New Mexico 87131, USA 102The Rockefeller University, New York, New York 10065, USA 103Rutgers University, Piscataway, New Jersey 08855, USA 104Princeton University, Princeton, New Jersey 08544, USA 105State University of New York, Buffalo, New York 14260, USA 106Columbia University, New York, New York 10027, USA 107University of Rochester, Rochester, New York 14627, USA aVisitor from Augustana College, Sioux Falls, SD, USA. bVisitor from Universidad de Oviedo, E-33007 Oviedo, Spain. cVisitor from Northwestern University, Evanston, IL 60208, USA. dVisitor from CNRS-IN2P3, Paris, F-75205, France. eVisitor from The University of Liverpool, Liverpool, United Kingdom. fVisitor from Universidad Iberoamericana, Mexico D.F., Mexico. gVisitor from ETH, 8092 Zurich, Switzerland. hVisitor from CERN,CH-1211 Geneva, Switzerland. iVisitor from Queen Mary, University of London, London, E1 4NS, United Kingdom. jVisitor from University of Melbourne, Victoria 3010, Australia. kVisitor from National Research Nuclear University, Moscow, Russia. lVisitor from University of California Santa Cruz, Santa Cruz, CA 95064, USA. mVisitor from Yarmouk University, Irbid 211-63, Jordan. nVisitor from Muons, Inc., Batavia, IL 60510, USA. oVisitor from UPIITA-IPN, Mexico City, Mexico. pVisitor from University of Cyprus, Nicosia CY-1678, Cyprus. qVisitor from DESY, Hamburg, Germany. rVisitor from SLAC, Menlo Park, CA, USA. sVisitor from University College London, London, United Kingdom. tVisitor from Cornell University, Ithaca, NY 14853, USA. uVisitor from Kansas State University, Manhattan, KS 66506, USA. vVisitor from Kinki University, Higashi-Osaka City, Japan 577-8502. wDeceased. xVisitor from University of California Santa Barbara, Santa Barbara, CA 93106, USA. yVisitor from University of Notre Dame, Notre Dame, IN 46556, USA. zVisitor from Korea University, Seoul, 136-713, Korea. aaVisitor from Texas Tech University, Lubbock, TX 79609, USA. bbVisitor from Centro de Investigacion en Computacion-IPN, Mexico City, Mexico. ccVisitor from Institute of Physics, Academy of Sciences of the Czech Republic, Czech Republic. ddVisitor from Istituto Nazionale di Fisica Nucleare, Sezione di Cagliari, 09042 Monserrato (Cagliari), Italy. eeVisitor from University College Dublin, Dublin 4, Ireland. ffVisitor from University of Iowa, Iowa City, IA 52242, USA. ggVisitor from ECFM, Universidad Autonoma de Sinaloa, Culiacán, Mexico. hhVisitor from Universidad Tecnica Federico Santa Maria, 110v Valparaiso, Chile. iiVisitor from Office of Science, U.S. Department of Energy, Washington, DC 20585, USA. jjVisitor from Universidade Estadual Paulista, São Paulo, Brazil. kkVisitor from Nagasaki Institute of Applied Science, Nagasaki, Japan. llVisitor from University of California Irvine, Irvine, CA 92697, USA. mmVisitor from University of Manchester, Manchester M13 9PL, United Kingdom. nnVisitor from University of Fukui, Fukui City, Fukui Prefecture, Japan 910-0017. ooDeceased. COMBINATION OF CDF AND D0 MEASUREMENTS OF . . . PHYSICAL REVIEW D 85, 071106(R) (2012) RAPID COMMUNICATIONS 071106-5 108State University of New York, Stony Brook, New York 11794, USA 109Brookhaven National Laboratory, Upton, New York 11973, USA 110Duke University, Durham, North Carolina 27708, USA 111The Ohio State University, Columbus, Ohio 43210, USA 112Langston University, Langston, Oklahoma 73050, USA 113University of Oklahoma, Norman, Oklahoma 73019, USA 114Oklahoma State University, Stillwater, Oklahoma 74078, USA 115Carnegie Mellon University, Pittsburgh, Pennsylvania 15213, USA 116University of Pittsburgh, Pittsburgh, Pennsylvania 15260, USA 117University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA 118Brown University, Providence, Rhode Island 02912, USA 119University of Texas, Arlington, Texas 76019, USA 120Texas A&M University, College Station, Texas 77843, USA 121Southern Methodist University, Dallas, Texas 75275, USA 122Rice University, Houston, Texas 77005, USA 123Baylor University, Waco, Texas 76798, USA 124University of Virginia, Charlottesville, Virginia 22904, USA 125University of Washington, Seattle, Washington 98195, USA 126University of Wisconsin, Madison, Wisconsin 53706, USA (Received 24 February 2012; published 27 April 2012) We report the combination of recent measurements of the helicity of theW boson from top quark decay by the CDF and D0 collaborations, based on data samples corresponding to integrated luminosities of 2:7–5:4 fb�1 of p �p collisions collected during Run II of the Fermilab Tevatron collider. Combining measurements that simultaneously determine the fractions of W bosons with longitudinal (f0) and right- handed (fþ) helicities, we find f0 ¼ 0:722� 0:081½�0:062ðstatÞ � 0:052ðsystÞ� and fþ ¼ �0:033� 0:046½�0:034ðstatÞ � 0:031ðsystÞ�. Combining measurements where one of the helicity fractions is fixed to the value expected in the standard model, we find f0 ¼ 0:682� 0:057½�0:035ðstatÞ � 0:046ðsystÞ� for fixed fþ and fþ ¼ �0:015� 0:035½�0:018ðstatÞ � 0:030ðsystÞ� for fixed f0. The results are consistent with standard model expectations. DOI: 10.1103/PhysRevD.85.071106 PACS numbers: 14.65.Ha, 12.15.Ji, 12.38.Qk, 14.70.Fm I. INTRODUCTION The study of the properties of the top quark is one of the major topics of the Tevatron proton-antiproton collider program at Fermilab. Using data samples 2 orders of magnitude larger than were available when the top quark was first observed [1], the CDF and D0 collaborations have investigated many properties of the top quark, including the helicity of the W bosons produced in the decays t ! Wb. The on-shell W bosons from top quark decays can have three possible helicity states, and we denote the fractions of Wþ bosons produced in these states as f0 (longitudinal), f� (left-handed), and fþ (right-handed). In the standard model (SM), the top quark decays via the V-A weak charged-current interaction, which strongly sup- presses right-handed Wþ bosons or left-handed W� bo- sons. The SM expectation for the helicity fractions depends upon the masses of the top quark (mt) and the W boson (MW). For the world average values mt ¼ 173:3� 1:1 GeV=c2 [2] and MW ¼ 80:399� 0:023 GeV=c2 [3], the expected SM values are f0 ¼ 0:688� 0:004, f� ¼ 0:310� 0:004, and fþ ¼ 0:0017� 0:0001 [4]. A measurement that deviates significantly from these expectations would provide strong evidence of phys- ics beyond the SM, indicating either a departure from the expected V-A structure of the tWb vertex or the presence of a non-SM contribution to the t�t candidate sample. We report the combination of recent measurements of f0 and fþ from data recorded at the Tevatron p �p collider by the CDF and D0 collaborations. The measurements are com- bined accounting for statistical and systematic correlations using the method of Refs. [5,6]. II. INPUT MEASUREMENTS The inputs to the combination are the f0 and fþ values extracted from 2:7 fb�1 of CDF data in the leptonþ jets (t�t ! WþW�b �b ! ‘�q �q0b �b) channel [7] and 5:1 fb�1 of CDF data in the dilepton (t�t ! WþW�b �b ! ‘�‘0�b �b) channel [8] (where ‘ and ‘0 represent an electron or a muon), and from 5:4 fb�1 of D0 data for leptonþ jets and dilepton events analyzed jointly [9]. All of these measurements use data collected during Run II of the Tevatron. Assuming f� þ f0 þ fþ ¼ 1, two types of mea- surements are performed: (i) a model-independent ap- proach where f0 and fþ are determined simultaneously, and (ii) a model-dependent approach where f0 ðfþÞ is fixed to its SM value, and fþ ðf0Þ is measured. The model- T. AALTONEN et al. PHYSICAL REVIEW D 85, 071106(R) (2012) RAPID COMMUNICATIONS 071106-6 http://dx.doi.org/10.1103/PhysRevD.85.071106 independent and model-dependent approaches are referred to as ‘‘2D’’ and ‘‘1D,’’ respectively. We label the input measurements as follows: (i) CDF’s measurements of f0 and fþ in the leptonþ jets channel are labeled as fnD;‘þj 0;CDF and fnD;‘þj þ;CDF , respectively. (ii) CDF’s measurements of f0 and fþ in the dilepton channel are labeled asfnD;‘‘0;CDF andf nD;‘‘ þ;CDF, respectively. (iii) D0’s measurements of f0 and fþ, which use both the leptonþ jets and dilepton channels, are labeled as fnD0;D0 and fnDþ;D0, respectively. Here n ¼ 1 for 1D measurements and n ¼ 2 for 2D measurements. The fnD;‘þj 0ðþÞ;CDF measurements [7] use the ‘‘matrix ele- ment’’ method described in Ref. [10], where the distribu- tions of themomenta ofmeasured jets and leptons as well as the missing transverse energy 6ET are compared to the expectations for leading-order signal and background ma- trix elements, convoluted with the detector response to jets and leptons. The t�t matrix elements are computed as a function of theW boson helicity fractions to determine the values of f0 and fþ that are most consistent with the data. The fnD;‘‘0ðþÞ;CDF and fnD0ðþÞ;D0 measurements are based on the distribution of the helicity angle �? for each top quark decay, where �? is the angle in the W boson rest frame between the direction opposite to the top quark and the direction of the down-type fermion (charged lepton or down-type quark) from the decay of the W boson. The probability distribution in cos�? can be written in terms of the helicity fractions as follows: !ðcos�?Þ / 2ð1� cos2�?Þf0 þ ð1� cos�?Þ2f� þ ð1þ cos�?Þ2fþ: (1) The momentum of the neutrino required to determine �? is reconstructed in the leptonþ jets channel through a con- strained kinematic fit of each event to the t�t hypothesis, while for the dilepton channel �? is obtained through an algebraic solution of the kinematics. The distributions in cos�? are compared to the expectations from background and t�t Monte Carlo (MC) simulated events, with different admixtures of helicity fractions, to determine f0 and fþ. CDF and D0 treat the top quark mass dependence of the measured helicity fractions differently. CDF assumes a value of mt ¼ 175 GeV=c2 when reporting central values and includes a description of how the values change as a function ofmt. D0 assumes a value ofmt ¼ 172:5 GeV=c2 and assigns a systematic uncertainty to cover the mt de- pendence of the result. This uncertainty corresponds to a 1:4 GeV=c2 uncertainty on mt, accounting for both the difference between D0’s assumedmt and the world average value and the uncertainty on the world average value [2]. To facilitate the combination of results, the CDF helicity fractions are shifted to mt of 172:5 GeV=c2, and an uncer- tainty is assigned to account for the 1:4 GeV=c2 uncer- tainty on mt. CDF and D0 also use slightly different MW values in their measurements (80:450 GeV=c2 for CDF and 80:419 GeV=c2 for D0), but this difference changes the expected helicity fractions only by � 10�4. The input measurements are summarized in Table I. III. CATEGORIES OF UNCERTAINTY The uncertainties on the individual measurements are grouped into categories so that the correlations can be treated properly in the combination. The categories are specified as follows: (i) STA is the statistical uncertainty. In each 2D input measurement, there is a strong anticorrelation between the values of f0 and fþ. The correlation coefficients are determined from the covariance matrix that is calculated during the simultaneous fit for f0 and fþ to be �0:8 in the D0 measurement, �0:6 in CDF’s leptonþ jets, and �0:9 in CDF’s dilepton measurement. (ii) JES is the uncertainty on the jet energy scale. This uncertainty can arise from theoretical uncertainties on the properties of jets, such as the models for TABLE I. Summary of the W boson helicity measurements used in the combination of results. The CDF measurements have been shifted from their published values to reflect a change in the assumed top quark mass from 175 to 172:5 GeV=c2. The first uncertainty in brackets below is statistical and the second is systematic. CDF leptonþ jets, 2:7 fb�1 [7] f2D;‘þj 0;CDF f0 ¼ 0:903� 0:123½�0:106� 0:063� f2D;‘þj þ;CDF fþ ¼ �0:195� 0:090½�0:067� 0:060� f1D;‘þj 0;CDF f0 ¼ 0:674� 0:081½�0:069� 0:042� f1D;‘þj þ;CDF fþ ¼ �0:044� 0:053½�0:019� 0:050� CDF dilepton, 5:1 fb�1[8] f2D;‘‘0;CDF f0 ¼ 0:702� 0:186½�0:175� 0:062� f2D;‘‘þ;CDF fþ ¼ �0:085� 0:096½�0:089� 0:035� f1D;‘‘0;CDF f0 ¼ 0:556� 0:106½�0:088� 0:060� f1D;‘‘þ;CDF fþ ¼ �0:089� 0:052½�0:041� 0:032� D0, leptonþ jets and dilepton, 5:4 fb�1 [9] f2D0;D0 f0 ¼ 0:669� 0:102½�0:078� 0:065� f2Dþ;D0 fþ ¼ 0:023� 0:053½�0:041� 0:034� f1D0;D0 f0 ¼ 0:708� 0:065½�0:044� 0:048� f1Dþ;D0 fþ ¼ 0:010� 0:037½�0:022� 0:030� COMBINATION OF CDF AND D0 MEASUREMENTS OF . . . PHYSICAL REVIEW D 85, 071106(R) (2012) RAPID COMMUNICATIONS 071106-7 gluon radiation and the fragmentation of b quarks (assessed by comparing the default model [11] to an alternative version [12]), and from uncertainties in the calorimeter response. We assume that the theo- retical uncertainties common to CDF and D0 domi- nate, and therefore take this uncertainty as fully correlated between CDF and D0. Details of the jet energy calibration in CDF and D0 can be found in Refs. [13,14], respectively. (iii) SIG is the uncertainty on the modeling of t�t pro- duction and decay and has several components. The effect of uncertainties on the parton distribution functions (PDFs) is estimated using the 2� 20 uncertainty sets provided for the CTEQ6M [15] PDFs. The uncertainty on the modeling of initial- and final-state gluon radiation is assessed by vary- ing the MC parameters for these processes. Uncertainties from modeling hadron showers are estimated by comparing the expectations from PYTHIA [17] and HERWIG [18]. In addition, D0 estimates the potential impact of next-to-leading order (NLO) effects by comparing the leading- order generators (ALPGEN [16], PYTHIA, and HERWIG) with the NLO generator MC@NLO [19], and the uncertainty from color reconnection [20] by comparing PYTHIA models with color reconnection turned on and off. These additional terms increase the t�t modeling uncertainty by 33% relative to the value that would be determined using only the components considered in the CDF analyses. Signal modeling uncertainties impact the CDF and D0 results in the same manner and therefore are taken as fully correlated among input measurements. (iv) BGD is the uncertainty on the modeling of the background. The procedures used to estimate this uncertainty differ for the separate analyses. In CDF’s dilepton measurement, the contribution of each background source is varied within its uncer- tainty and the resulting effect on the cos�? distri- bution is used to gauge the effect on the measured helicity fractions. In the CDF leptonþ jets analy- sis, the change in the result when the background is assumed to come from only one source (e.g. only W þ b �b production or only multijet production), rather than from the expected mixture of sources, is taken as the uncertainty due to the background shape. The uncertainty on the background yield is evaluated by varying the assumed signal-to- background ratio. In the D0 measurement, the cos�? distributions in data and in the background model are compared in a background-dominated sideband region. The background model in the signal region is then reweighted to reflect any dif- ferences observed in the background-dominated region, and the resulting changes in the measured helicity fractions are taken as their systematic un- certainties. The correlations among the background model uncertainties in the input measurements are not known, but are presumably large because of the substantial contribution ofW=Zþ jets events to the background in each measurement. We therefore TABLE II. Relationship between the individual systematic uncertainties on the input measurements [7–9] and the categories of uncertainty used for the combination. Individual measurement uncertainties Uncertainty category CDF leptonþ jets CDF dilepton D0 leptonþ jets and dilepton JES Jet energy scale Jet energy scale Jet energy scale b fragmentation SIG Initial state radiation or final state radiation Generators t�t model PDF Initial state radiation or final state radiation PDF Parton shower PDF BGD Background Background shape Background model Heavy flavor fraction MTD Method-related Template statistics Template statistics Analysis consistency MTOP Top quark mass Top quark mass Top quark mass DET Jet energy resolution Jet identification Muon identification Muon trigger MHI Instantaneous luminosity T. AALTONEN et al. PHYSICAL REVIEW D 85, 071106(R) (2012) RAPID COMMUNICATIONS 071106-8 treat this uncertainty as fully correlated between CDF and D0, and also between measurements using dilepton and leptonþ jets events. (v) MTD are uncertainties that are specific to a given analysis method. Effects such as the limitations from the statistics of the MC and any offsets ob- served in self-consistency tests of the analysis are included in this category. These uncertainties are fully anticorrelated for 2D measurements of f0 and fþ within a given analysis, but not between different analyses. (vi) MTOP is the uncertainty due to mt and is fully correlated between all measurements. (vii) DET are uncertainties due to the response of the CDF and D0 detectors. The effects considered include uncertainty in jet energy resolution, lepton identification efficiency, and trigger efficiency. These uncertainties are found to be negligible in the CDF measurements, but are larger in the D0 measurements due to discrepancies observed in muon distributions between data control samples and MC. While the cause of these discrepancies was subsequently understood and resolved, D0 assigns a systematic uncertainty to cover the effect rather than reanalyzing the data. (viii) MHI is the uncertainty due to multiple hadronic (p �p) interactions in a single bunch crossing. This uncertainty pertains only to the CDF dilepton measurement. In D0’s measurements the distribu- tion in instantaneous luminosity for the simulated events is reweighted to match that in data, thereby accounting for the impact of multiple interactions. In CDF’s leptonþ jets measurement this uncer- tainty is found to be negligible. The relationships between the uncertainties reported in individual measurements [7–9] and the above categories are given in Table II, and the values of the uncertainties from each input measurement are given in Table III. IV. COMBINATION PROCEDURE The results are combined to obtain the best linear un- biased estimators of the correlated observables f0 and fþ [5]. The method uses all the measurements and their co- variance matrix M, where M is the sum of the covariance matrices for each category of uncertainty (for the 1D measurements, only the submatrices corresponding to the helicity fraction that is varied are relevant): M ¼ MSTA þMJES þMSIG þMBGD þMMTD þMMTOP þMDET þMMHI: (2) The correlation coefficients assumed when populating the covariance matrices for each category of uncertainty are summarized in the above discussion of systematic uncer- tainties. When there are correlations in systematic uncer- tainties between measurements of f0 and fþ, the correlation coefficients are taken to be �1, reflecting the large negative statistical correlations observed between measurements of f0 and fþ within a given analysis. V. RESULTS The result of the combination of the 2D measurements is f0 ¼ 0:722� 0:081 ½�0:062ðstatÞ � 0:052ðsystÞ�; fþ ¼ �0:033� 0:046 ½�0:034ðstatÞ � 0:031ðsystÞ�: (3) The contribution from each category of systematic uncertainty is shown in Table IV. The combination has a �2 value of 8.86 for 4 degrees of freedom, corresponding to a p-value of 6% for consistency among the input TABLE III. Values of the uncertainties from each measurement that are used in the combi- nations. Measurement STA JES SIG BGD MTD MTOP DET MHI f2D;‘þj 0;CDF 0.106 0.004 0.038 0.042 0.024 0.011 0.000 0.000 f2D0;D0 0.078 0.011 0.039 0.032 0.022 0.009 0.031 0.000 f2D;‘‘0;CDF 0.175 0.002 0.050 0.023 0.028 0.005 0.000 0.013 f2D;‘þj þ;CDF 0.067 0.012 0.031 0.039 0.024 0.019 0.000 0.000 f2Dþ;D0 0.041 0.009 0.024 0.013 0.012 0.012 0.007 0.000 f2D;‘‘þ;CDF 0.089 0.020 0.022 0.010 0.014 0.005 0.000 0.002 f1D;‘þj 0;CDF 0.069 0.018 0.033 0.009 0.010 0.012 0.000 0.000 f1D0;D0 0.044 0.016 0.036 0.013 0.021 0.012 0.018 0.000 f1D;‘‘0;CDF 0.088 0.033 0.044 0.012 0.012 0.013 0.000 0.016 f1D;‘þj þ;CDF 0.019 0.017 0.024 0.038 0.005 0.015 0.000 0.000 f1Dþ;D0 0.022 0.012 0.021 0.008 0.008 0.010 0.010 0.000 f1D;‘‘þ;CDF 0.041 0.019 0.022 0.005 0.006 0.007 0.000 0.008 COMBINATION OF CDF AND D0 MEASUREMENTS OF . . . PHYSICAL REVIEW D 85, 071106(R) (2012) RAPID COMMUNICATIONS 071106-9 measurements. The combined values of f0 and fþ have a correlation coefficient of �0:86. Contours of constant �2 in the f0 and fþ plane are shown in Fig. 1. The SM values for the helicity fractions lie within the 68% C.L. contour of probability. Combining the 1D measurements yields: f0 ¼ 0:682� 0:057 ½�0:035ðstatÞ � 0:046ðsystÞ�; fþ ¼ �0:015� 0:035 ½�0:018ðstatÞ � 0:030ðsystÞ�: (4) The contribution of each category of systematic uncer- tainty is shown in Table IV. The combination for f0 (fþ) has a �2 of 2.12 (4.44) for 2 degrees of freedom, corre- sponding to a p-value of 35% (11%) for consistency among the input measurements. VI. SUMMARY We have combined measurements of the helicity of W bosons arising from top quark decay in t�t events from the CDF and D0 collaborations, providing the most precise measurements of f0 and fþ to date. The results are con- sistent with expectations from the SM and provide no indication of new physics in the tWb coupling or of the presence of a non-SM source of events in the selected sample. ACKNOWLEDGMENTS We thank the staffs at Fermilab and collaborating insti- tutions, and acknowledge support from the DOE and NSF (USA); CEA and CNRS/IN2P3 (France); FASI, Rosatom and RFBR (Russia); CNPq, FAPERJ, FAPESP, and FUNDUNESP (Brazil); DAE and DST (India); INFN (Italy); Ministry of Education, Culture, Sports, Science, and Technology (Japan); Colciencias (Colombia); CONACyT (Mexico); World Class University Program, National Research Foundation, NRF (Korea); CONICET and UBACyT (Argentina); Australian Research Council (Australia); FOM (The Netherlands); STFC and the Royal Society (United Kingdom); MSMT and GACR (Czech Republic); CRC Program and NSERC (Canada); Academy of Finland (Finland); BMBF and DFG (Germany); SFI (Ireland); Slovak R&D Agency (Slovakia); Programa Consolider-Ingenio 2010 (Spain); Swedish Research Council (Sweden); Swiss National Science Foundation (Switzerland); NSC (Republic of China); CAS and CNSF (China); and the A. P. Sloan Foundation (USA). [1] F. Abe et al. (CDF Collaboration), Phys. Rev. Lett. 74, 2626 (1995); S. Abachi et al. (D0 Collaboration), Phys. Rev. Lett. 74, 2632 (1995). [2] Tevatron Electroweak Working Group, CDF Collaboration, and D0 Collaboration, arXiv:1007.3178. [3] K. Nakamura et al. (Particle Data Group), J. Phys. G 37, 075021 (2010). [4] A. Czarnecki, J. G. Körner, and J. H. Piclum, Phys. Rev. D 81, 111503 (2010). The helicity fractions for mt ¼ 173:3� 1:1 GeV=c2 and MW ¼ 80:399� 0:023 GeV=c2 TABLE IV. The contribution from each category of systematic uncertainty in the combined measurements. Category 2D combination 1D combination �f0 �fþ �f0 �fþ JES 0.007 0.012 0.018 0.014 SIG 0.038 0.022 0.036 0.021 BGD 0.028 0.013 0.012 0.009 MTD 0.014 0.008 0.007 0.006 MTOP 0.007 0.010 0.012 0.010 DET 0.016 0.003 0.011 0.007 MHI 0.001 0.0004 0.002 0.002 +f -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0f 0.5 0.6 0.7 0.8 0.9 1 Combined result SM value CDF l+jets CDF dilepton DØ CDF + DØ combination L = 2.7 - 5.4 fb-1 68% and 95% C.L. contours FIG. 1 (color online). Contours of constant �2 for the combi- nation of the 2D helicity measurements. The ellipses indicate the 68% and 95% C.L. contours, the dot shows the best-fit value, and the star marks the expectation from the SM. The input measure- ments to the combination are represented by the open circle, square, and triangle, with error bars indicating the 1� uncertain- ties on f0 and fþ. Each of the input measurements uses a central value of mt ¼ 172:5 GeV=c2. T. AALTONEN et al. 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