arXiv:1605.02719v2 [hep-ex] 10 Jun 2016 Measurement of sin2 θeleffpt using e+e− pairs from γ∗/Z bosons produced in pp¯ collisions at a center-of-momentum energy of 1.96 TeV T. Aaltonen,21 S. Amerioll,39 D. Amidei,31 A. Anastassovw,15 A. Annovi,17 J. Antos,12 G. Apollinari,15 J.A. Appel,15 T. Arisawa,51 A. Artikov,13 J. Asaadi,47 W. Ashmanskas,15 B. Auerbach,2 A. Aurisano,47 F. Azfar,38 W. Badgett,15 T. Bae,25 A. Barbaro-Galtieri,26 V.E. Barnes,43 B.A. Barnett,23 P. Barriann,41 P. Bartos,12 M. Baucell,39 F. Bedeschi,41 S. Behari,15 G. Bellettinimm,41 J. Bellinger,53 D. Benjamin,14 A. Beretvas,15 A. Bhatti,45 K.R. Bland,5 B. Blumenfeld,23 A. Bocci,14 A. Bodek,44 D. Bortoletto,43 J. Boudreau,42 A. Boveia,11 L. Brigliadorikk,6 C. Bromberg,32 E. Brucken,21 J. Budagov,13 H.S. Budd,44 K. Burkett,15 G. Busettoll,39 P. Bussey,19 P. Buttimm,41 A. Buzatu,19 A. Calamba,10 S. Camarda,4 M. Campanelli,28 F. Canelliee,11 B. Carls,22 D. Carlsmith,53 R. Carosi,41 S. Carrillol,16 B. Casalj,9 M. Casarsa,48 A. Castrokk,6 P. Catastini,20 D. Cauzss,tt,48 V. Cavaliere,22 A. Cerrie,26 L. Cerritor,28 Y.C. Chen,1 M. Chertok,7 G. Chiarelli,41 G. Chlachidze,15 K. Cho,25 D. Chokheli,13 A. Clark,18 C. Clarke,52 M.E. Convery,15 J. Conway,7 M. Corboz,15 M. Cordelli,17 C.A. Cox,7 D.J. Cox,7 M. Cremonesi,41 D. Cruz,47 J. Cuevasy,9 R. Culbertson,15 N. d’Ascenzov,15 M. Dattahh,15 P. de Barbaro,44 L. Demortier,45 M. Deninno,6 M. D’Erricoll,39 F. Devoto,21 A. Di Cantomm,41 B. Di Ruzzap,15 J.R. Dittmann,5 S. Donatimm,41 M. D’Onofrio,27 M. Dorigouu,48 A. Driuttiss,tt,48 K. Ebina,51 R. Edgar,31 R. Erbacher,7 S. Errede,22 B. Esham,22 S. Farrington,38 J.P. Fern´andez Ramos,29 R. Field,16 G. Flanagant,15 R. Forrest,7 M. Franklin,20 J.C. Freeman,15 H. Frisch,11 Y. Funakoshi,51 C. Gallonimm,41 A.F. Garfinkel,43 P. Garosinn,41 H. Gerberich,22 E. Gerchtein,15 S. Giagu,46 V. Giakoumopoulou,3 K. Gibson,42 C.M. Ginsburg,15 N. Giokaris,3 P. Giromini,17 V. Glagolev,13 D. Glenzinski,15 M. Gold,34 D. Goldin,47 A. Golossanov,15 G. Gomez,9 G. Gomez-Ceballos,30 M. Goncharov,30 O. Gonz´alez L´opez,29 I. Gorelov,34 A.T. Goshaw,14 K. Goulianos,45 E. Gramellini,6 C. Grosso-Pilcher,11 J. Guimaraes da Costa,20 S.R. Hahn,15 J.Y. Han,44 F. Happacher,17 K. Hara,49 M. Hare,50 R.F. Harr,52 T. Harrington-Taberm,15 K. Hatakeyama,5 C. Hays,38 J. Heinrich,40 M. Herndon,53 A. Hocker,15 Z. Hong,47 W. Hopkinsf ,15 S. Hou,1 R.E. Hughes,35 U. Husemann,54 M. Husseincc,32 J. Huston,32 G. Introzzipp,qq,41 M. Iorirr,46 A. Ivanovo,7 E. James,15 D. Jang,10 B. Jayatilaka,15 E.J. Jeon,25 S. Jindariani,15 M. Jones,43 K.K. Joo,25 S.Y. Jun,10 T.R. Junk,15 M. Kambeitz,24 T. Kamon,25, 47 P.E. Karchin,52 A. Kasmi,5 Y. Katon,37 W. Ketchumii,11 J. Keung,40 B. Kilminsteree,15 D.H. Kim,25 H.S. Kimbb,15 J.E. Kim,25 M.J. Kim,17 S.H. Kim,49 S.B. Kim,25 Y.J. Kim,25 Y.K. Kim,11 N. Kimura,51 M. Kirby,15 K. Kondo,51, ∗ D.J. Kong,25 J. Konigsberg,16 A.V. Kotwal,14 M. Kreps,24 J. Kroll,40 M. Kruse,14 T. Kuhr,24 M. Kurata,49 A.T. Laasanen,43 S. Lammel,15 M. Lancaster,28 K. Lannonx,35 G. Latinonn,41 H.S. Lee,25 J.S. Lee,25 S. Leo,22 S. Leone,41 J.D. Lewis,15 A. Limosanis,14 E. Lipeles,40 A. Listera,18 Q. Liu,43 T. Liu,15 S. Lockwitz,54 A. Loginov,54, ∗ D. Lucchesill,39 A. Luc`a,17 J. Lueck,24 P. Lujan,26 P. Lukens,15 G. Lungu,45 J. Lys,26 R. Lysakd,12 R. Madrak,15 P. Maestronn,41 S. Malik,45 G. Mancab,27 A. Manousakis-Katsikakis,3 L. Marchesejj,6 F. Margaroli,46 P. Marinooo,41 K. Matera,22 M.E. Mattson,52 A. Mazzacane,15 P. Mazzanti,6 R. McNultyi,27 A. Mehta,27 P. Mehtala,21 C. Mesropian,45 T. Miao,15 D. Mietlicki,31 A. Mitra,1 H. Miyake,49 S. Moed,15 N. Moggi,6 C.S. Moonz,15 R. Mooreff,gg,15 M.J. Morellooo,41 A. Mukherjee,15 Th. Muller,24 P. Murat,15 M. Mussinikk,6 J. Nachtmanm,15 Y. Nagai,49 J. Naganoma,51 I. Nakano,36 A. Napier,50 J. Nett,47 T. Nigmanov,42 L. Nodulman,2 S.Y. Noh,25 O. Norniella,22 L. Oakes,38 S.H. Oh,14 Y.D. Oh,25 T. Okusawa,37 R. Orava,21 L. Ortolan,4 C. Pagliarone,48 E. Palenciae,9 P. Palni,34 V. Papadimitriou,15 W. Parker,53 G. Paulettass,tt,48 M. Paulini,10 C. Paus,30 T.J. Phillips,14 G. Piacentinoq,15 E. Pianori,40 J. Pilot,7 K. Pitts,22 C. Plager,8 L. Pondrom,53 S. Poprockif ,15 K. Potamianos,26 A. Pranko,26 F. Prokoshinaa,13 F. Ptohosg,17 G. Punzimm,41 I. Redondo Fern´andez,29 P. Renton,38 M. Rescigno,46 F. Rimondi,6, ∗ L. Ristori,41, 15 A. Robson,19 T. Rodriguez,40 S. Rollih,50 M. Ronzanimm,41 R. Roser,15 J.L. Rosner,11 F. Ruffininn,41 A. Ruiz,9 J. Russ,10 V. Rusu,15 W.K. Sakumoto,44 Y. Sakurai,51 L. Santiss,tt,48 K. Sato,49 V. Savelievv,15 A. Savoy-Navarroz,15 P. Schlabach,15 E.E. Schmidt,15 T. Schwarz,31 L. Scodellaro,9 F. Scuri,41 S. Seidel,34 Y. Seiya,37 A. Semenov,13 F. Sforzamm,41 S.Z. Shalhout,7 T. Shears,27 P.F. Shepard,42 M. Shimojimau,49 M. Shochet,11 I. Shreyber-Tecker,33 A. Simonenko,13 K. Sliwa,50 J.R. Smith,7 F.D. Snider,15 H. Song,42 V. Sorin,4 R. St. Denis,19, ∗ M. Stancari,15 D. Stentzw,15 J. Strologas,34 Y. Sudo,49 A. Sukhanov,15 I. Suslov,13 K. Takemasa,49 Y. Takeuchi,49 J. Tang,11 M. Tecchio,31 P.K. Teng,1 J. Thomf ,15 E. Thomson,40 V. Thukral,47 D. Toback,47 S. Tokar,12 K. Tollefson,32 T. Tomura,49 D. Tonellie,15 S. Torre,17 D. Torretta,15 P. Totaro,39 M. Trovatooo,41 F. Ukegawa,49 S. Uozumi,25 F. V´azquezl,16 G. Velev,15 C. Vellidis,15 C. Vernierioo,41 M. Vidal,43 R. Vilar,9 J. Viz´andd,9 M. Vogel,34 G. Volpi,17 P. Wagner,40 R. Wallnyj,15 2 S.M. Wang,1 D. Waters,28 W.C. Wester III,15 D. Whitesonc,40 A.B. Wicklund,2 S. Wilbur,7 H.H. Williams,40 J.S. Wilson,31 P. Wilson,15 B.L. Winer,35 P. Wittichf ,15 S. Wolbers,15 H. Wolfe,35 T. Wright,31 X. Wu,18 Z. Wu,5 K. Yamamoto,37 D. Yamato,37 T. Yang,15 U.K. Yang,25 Y.C. Yang,25 W.-M. Yao,26 G.P. Yeh,15 K. Yim,15 J. Yoh,15 K. Yorita,51 T. Yoshidak,37 G.B. Yu,14 I. Yu,25 A.M. Zanetti,48 Y. Zeng,14 C. Zhou,14 and S. Zucchellikk6 (CDF Collaboration)† 1Institute of Physics, Academia Sinica, Taipei, Taiwan 11529, Republic of China 2Argonne National Laboratory, Argonne, Illinois 60439, USA 3University of Athens, 157 71 Athens, Greece 4Institut de Fisica d’Altes Energies, ICREA, Universitat Autonoma de Barcelona, E-08193, Bellaterra (Barcelona), Spain 5Baylor University, Waco, Texas 76798, USA 6Istituto Nazionale di Fisica Nucleare Bologna, kkUniversity of Bologna, I-40127 Bologna, Italy 7University of California, Davis, Davis, California 95616, USA 8University of California, Los Angeles, Los Angeles, California 90024, USA 9Instituto de Fisica de Cantabria, CSIC-University of Cantabria, 39005 Santander, Spain 10Carnegie Mellon University, Pittsburgh, Pennsylvania 15213, USA 11Enrico Fermi Institute, University of Chicago, Chicago, Illinois 60637, USA 12Comenius University, 842 48 Bratislava, Slovakia; Institute of Experimental Physics, 040 01 Kosice, Slovakia 13Joint Institute for Nuclear Research, RU-141980 Dubna, Russia 14Duke University, Durham, North Carolina 27708, USA 15Fermi National Accelerator Laboratory, Batavia, Illinois 60510, USA 16University of Florida, Gainesville, Florida 32611, USA 17Laboratori Nazionali di Frascati, Istituto Nazionale di Fisica Nucleare, I-00044 Frascati, Italy 18University of Geneva, CH-1211 Geneva 4, Switzerland 19Glasgow University, Glasgow G12 8QQ, United Kingdom 20Harvard University, Cambridge, Massachusetts 02138, USA 21Division of High Energy Physics, Department of Physics, University of Helsinki, FIN-00014, Helsinki, Finland; Helsinki Institute of Physics, FIN-00014, Helsinki, Finland 22University of Illinois, Urbana, Illinois 61801, USA 23The Johns Hopkins University, Baltimore, Maryland 21218, USA 24Institut fu¨r Experimentelle Kernphysik, Karlsruhe Institute of Technology, D-76131 Karlsruhe, Germany 25Center 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; Chonbuk National University, Jeonju 561-756, Korea; Ewha Womans University, Seoul, 120-750, Korea 26Ernest Orlando Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA 27University of Liverpool, Liverpool L69 7ZE, United Kingdom 28University College London, London WC1E 6BT, United Kingdom 29Centro de Investigaciones Energeticas Medioambientales y Tecnologicas, E-28040 Madrid, Spain 30Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA 31University of Michigan, Ann Arbor, Michigan 48109, USA 32Michigan State University, East Lansing, Michigan 48824, USA 33Institution for Theoretical and Experimental Physics, ITEP, Moscow 117259, Russia 34University of New Mexico, Albuquerque, New Mexico 87131, USA 35The Ohio State University, Columbus, Ohio 43210, USA 36Okayama University, Okayama 700-8530, Japan 37Osaka City University, Osaka 558-8585, Japan 38University of Oxford, Oxford OX1 3RH, United Kingdom 39Istituto Nazionale di Fisica Nucleare, Sezione di Padova, llUniversity of Padova, I-35131 Padova, Italy 40University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA 41Istituto Nazionale di Fisica Nucleare Pisa, mmUniversity of Pisa, nnUniversity of Siena, ooScuola Normale Superiore, I-56127 Pisa, Italy, ppINFN Pavia, I-27100 Pavia, Italy, qqUniversity of Pavia, I-27100 Pavia, Italy 42University of Pittsburgh, Pittsburgh, Pennsylvania 15260, USA 43Purdue University, West Lafayette, Indiana 47907, USA 44University of Rochester, Rochester, New York 14627, USA 45The Rockefeller University, New York, New York 10065, USA 46Istituto Nazionale di Fisica Nucleare, Sezione di Roma 1, rrSapienza Universit`a di Roma, I-00185 Roma, Italy 47Mitchell Institute for Fundamental Physics and Astronomy, Texas A&M University, College Station, Texas 77843, USA 48Istituto Nazionale di Fisica Nucleare Trieste, ssGruppo Collegato di Udine, ttUniversity of Udine, I-33100 Udine, Italy, uuUniversity of Trieste, I-34127 Trieste, Italy 49University of Tsukuba, Tsukuba, Ibaraki 305, Japan 50Tufts University, Medford, Massachusetts 02155, USA 51Waseda University, Tokyo 169, Japan 52Wayne State University, Detroit, Michigan 48201, USA 53University of Wisconsin-Madison, Madison, Wisconsin 53706, USA 54Yale University, New Haven, Connecticut 06520, USA (Dated: June 13, 2016) At the Fermilab Tevatron proton-antiproton (pp¯) collider, Drell-Yan lepton pairs are produced in the process pp¯ → e+e− +X through an intermediate γ∗/Z boson. The forward-backward asymmetry in the polar-angle distribution of the e− as a function of the e+e−-pair mass is used to obtain sin2 θeleffpt, the measurement effective leptonic determination sample, recorded by the Collider of the electroweak-mixing parameter sin2 θW . Detector at Fermilab (CDF), corresponds to 9.4 The fb−1 of integrated luminosity from pp¯ collisions at a center-of-momentum energy of 1.96 TeV, and is the full CDF Run II data set. The value of sin2 θeleffpt is found to be 0.23248 ± 0.00053. The combination with the previous CDF measurement based on µ+µ− pairs yields sin2 θeleffpt = 0.23221 ± 0.00046. This result, when interpreted within the specified context of the standard model assuming sin2 θW = 1 − MW2 /MZ2 and that the W - and Z-boson masses are on-shell, yields sin2 θW = 0.22400 ± 0.00045, or equivalently a W -boson mass of 80.328 ± 0.024 GeV/c2. PACS numbers: 12.15.Lk, 13.85.Qk, 14.70.Hp 3 I. INTRODUCTION ∗ Deceased † With visitors from aUniversity of British Columbia, Vancou- ver, BC V6T 1Z1, Canada, bIstituto Nazionale di Fisica Nucleare, Sezione di Cagliari, 09042 Monserrato (Cagliari), Italy, cUniversity of California Irvine, Irvine, CA 92697, USA, dInstitute of Physics, Academy of Sciences of the Czech Republic, 182 21, Czech Republic, eCERN, CH-1211 Geneva, Switzerland, f Cornell University, Ithaca, NY 14853, USA, gUniversity of Cyprus, Nicosia CY-1678, Cyprus, hOffice of Science, U.S. Department of Energy, Washington, DC 20585, USA, iUniversity College Dublin, Dublin 4, Ireland, j ETH, 8092 Zu¨rich, Switzerland, kUniversity of Fukui, Fukui City, Fukui Prefecture, Japan 910-0017, lUniversidad Iberoamericana, Lomas de Santa Fe, M´exico, C.P. 01219, Distrito Federal, mUniversity of Iowa, Iowa City, IA 52242, USA, nKinki University, Higashi-Osaka City, Japan 577-8502, oKansas State University, Manhattan, KS 66506, USA, pBrookhaven National Laboratory, Upton, NY 11973, USA, qIstituto Nazionale di Fisica Nucleare, Sezione di Lecce, Via Arnesano, I-73100 Lecce, Italy, rQueen Mary, University of London, London, E1 4NS, United Kingdom, sUniversity of Melbourne, Victoria 3010, Australia, tMuons, Inc., Batavia, IL 60510, USA, uNagasaki Institute of Applied Science, Nagasaki 851-0193, Japan, vNational Research Nuclear University, Moscow 115409, Russia, wNorthwestern University, Evanston, IL 60208, USA, xUniversity of Notre Dame, Notre Dame, IN 46556, USA, yUniversidad de Oviedo, E-33007 Oviedo, Spain, zCNRS-IN2P3, Paris, F-75205 France, aaUniversidad Tecnica Federico Santa Maria, 110v Valparaiso, Chile, bbSejong University, Seoul 143-747, Korea, ccThe University of Jordan, Amman 11942, Jordan, ddUniversite catholique de Louvain, 1348 Louvain-La-Neuve, Belgium, eeUniversity of Zu¨rich, 8006 Zu¨rich, Switzerland, ff Massachusetts General Hospital, Boston, MA 02114 USA, ggHarvard Medical School, Boston, MA 02114 USA, hhHampton University, Hampton, VA 23668, USA, iiLos Alamos National Laboratory, Los Alamos, NM 87544, USA, jj Universita` degli Studi di Napoli Federico II, I-80138 Napoli, Italy In this paper, the angular distribution of charged leptons ( ±) from the Drell-Yan [1] process is used to measure the electroweak-mixing parameter sin2 θW [2]. At the Fermilab Tevatron collider, Drell-Yan pairs are produced by the process pp¯ → + − + X, where the + − pair is produced through an intermediate γ∗/Z boson, and X is the final state associated with the production of the boson. In the standard model, the production of Drell-Yan lepton pairs at the Born level proceeds through two parton-level processes, qq¯ → γ∗ → + − and qq¯ → Z → + −, where the q and q¯ are the quark and antiquark, respec- tively, from the colliding hadrons. The virtual photon couples the vector currents of the incoming and outgoing fermions (f ), and the spacetime structure of a photonfermion interaction vertex is f¯|Qf γµ|f , where Qf , the strength of the coupling, is the fermion charge (in units of e), and |f is the spinor for fermion f . An interaction vertex of a fermion with a Z boson contains both vec- tor (V ) and axial-vector (A) current components, and its structure is f¯|gVf γµ + gAf γµγ5|f . The Born-level coupling strengths are gVf = T3f − 2Qf sin2 θW and gAf = T3f , where T3f is the third component of the fermion weak- isospin, which is T3f = 1 2 (− 1 2 ) for positively (nega- tively) charged fermions. At the Born level, and in all 4 orders of the on-shell renormalization scheme [3], the sin2 θW parameter is related to the W -boson mass MW and the Z-boson mass MZ by the relationship sin2 θW = 1 − MW2 /MZ2 . Radiative corrections alter the strength of the Born-level couplings into effective couplings. These effective couplings have been investigated at the Tevatron [4–7], at the LHC [8–10], and at LEP-1 and SLC [11, 12]. The on-shell sin2 θW coupling has been investigated with neutrino-nucleon collisions at the Tevatron [13] and with electron-proton collisions at HERA [14]. The effective sin2 θW coupling at the lepton vertex, dLeEnPo-t1edaansdsiSnL2Cθeleffep+t,eh−ascobleliednerascc[u11ra, t1e2ly].mTeahseurceodmabtintehde average of six individual measurements yields a value of 0.23149 ± 0.00016. However, there is tension between the two most precise individual measurements: the combined LEP-1 and SLD b-quark forward-backward asymmetry (A0F,Bb) yields sin2 θeleffpt = 0.23221 ± 0.00029, and the SLD left-right polarization asymmetry of Z-boson production (A ) yields sin2 θeleffpt = 0.23098 ± 0.00026. They differ by 3.2 standard deviations. The Drell-Yan process at hadron-hadron colliders is also sensitive to the sin2 θeleffpt coupling. of the forward-backward asymmetry in Measurements the − polar- angle distribution as a function of the lepton-pair invari- ant mass are used to extract the coupling. This paper presents a new measurement of the sin2 θeleffpt coupling and an inference of the sin2 θW parameter using a sample of e+e− pairs corresponding to an integrated pp¯ luminosity of 9.4 fb−1 collected at the Tevatron pp¯ collider. Inno- vative methods for the calibration of the electron energy and the measurement of the forward-backward asymme- try are used. Electroweak radiative corrections used for the extraction of sin2 θeleffpt and sin2 θW are derived from an approach used by LEP-1 and SLD. An outline of the paper follows. Section II provides an overview of the lepton angular distributions and the extraction of sin2 θeleffpt. Section III discusses quantum chromodynamics (QCD) calculations for the forward- backward asymmetry and the inclusion of electroweak radiative-correction form factors used in the analysis of high-energy e+e− collisions. The form factors are required for the determination of sin2 θW from the measurement of sin2 θeleffpt. Section IV describes the experimental apparatus. Section V reports on the selection of data. Section VI describes the simulation of the re- constructed data. Sections VII and VIII present the experimental calibrations and the measurement of the asymmetry, respectively, along with corresponding cor- rections to data and simulation. Section IX describes the method used to extract sin2 θeleffpt. Section X describes the systematic uncertainties. Section XI presents the results of this measurement using e+e− pairs and Sec. XII describes the combination of results from this measurement and a previous CDF measurement using µ+µ− pairs [6]. Finally, Sec. XIII presents the summary. Standard units are used for numerical values of particle masses and momenta, e.g., 40 GeV/c2 and 20 GeV/c, respectively, where c denotes the speed of light. Otherwise, natural units ( = c = 1) are used. II. LEPTON ANGULAR DISTRIBUTION The angular distribution of leptons from the Drell-Yan process in the rest frame of the boson is governed by the polarization state of the γ∗/Z boson. In amplitudes at higher order than tree level, initial-state QCD interactions of the colliding partons impart to the γ∗/Z boson a momentum component transverse to the collision axis, thus affecting the polarization states. In the laboratory frame, the pp¯ collision axis is the z axis, with the positive direction oriented along the direction of the proton. The transverse component of any vector, such as the momentum vector, is defined to be relative to that axis. The transverse component of vectors in other reference frames is defined to be relative to the z axes in those frames. For the description of the Drell-Yan process, the ra- pidity, transverse momentum, and mass of a particle are denoted as y, PT, and M , respectively. The energy and momentum of particles are denoted as E and P , re- spectively. In a given coordinate frame, the rapidity is y = 1 2 ln[ (E + Pz)/(E − Pz) ], where Pz is the component of the momentum vector along the z axis of the coordi- nate frame. The polar and azimuthal angles of the − direction in the rest frame of the boson are denoted as ϑ and ϕ, respectively. For this analysis, the ideal positive z axis coincides with the direction of the incoming quark so that the definition of ϑ parallels the definition used in e+e− collisions at LEP [11, 12]. This frame is approximated by the Collins-Soper (CS) rest frame [15] for pp¯ collisions. The rest frame is reached from the laboratory frame via two Lorentz boosts, first along the laboratory z-axis into a frame where the z component of the leptonpair momentum vector is zero, followed by a boost along the transverse component of the lepton-pair momentum vector. A view of the CS frame is shown in Fig. 1. The general structure of the Drell-Yan lepton angulardistribution in the boson rest frame consists of terms from nine helicity cross-sections that describe the polar- 5 PT +x --PB +z +zlab PB PA FIG. 1. Representation of the Collins-Soper coordinate axes (x, z) in the lepton-pair rest frame, along with the laboratory z axis (zlab). The three axes are in the plane formed by the proton (PA) and antiproton (PB) momentum vectors in the rest frame. The z axis is the angular bisector of PA and −PB. The y axis is along the direction of PB × PA, and the x axis is in the direction away from the transverse component of PA + PB. ization state of the boson, dN ∝ (1 + cos2 ϑ) + dΩ A0 1 2 (1 − 3 cos2 ϑ) + A1 sin 2ϑ cos ϕ + A2 1 2 sin2 ϑ cos 2ϕ + A3 sin ϑ cos ϕ + A4 cos ϑ + A5 sin2 ϑ sin 2ϕ + A6 sin 2ϑ sin ϕ + A7 sin ϑ sin ϕ , (1) where each term is relative to the cross section for un- polarized production integrated over the lepton angular distribution [16]. The coefficients A0−7 are functions of kinematic variables of the boson and vanish when the lepton-pair transverse momentum is zero, except for A4, which contributes to the tree-level QCD amplitude and generates the forward-backward − asymmetry in cos ϑ. Thus, at zero transverse momentum, the angular distribution reduces to the tree-level form 1+cos2 ϑ+A4 cos ϑ. The A4 coefficient is relatively uniform across the range of transverse momentum where the cross section is large (at values smaller than approximately 45 GeV/c), but slowly decreases for larger values of transverse momen- tum, where the cross section is very small. The A0 and A2 coefficients, corresponding to the longitudinal and transverse states of polarization, respectively, are the most significant and have been previously measured, along with A3 and A4 [17]. The A1 coefficient, from the interference between the longitudinal and transverse states of polarization, is small in the CS frame. The A5−7 coefficients appear at second order in the QCD strong coupling, αs, and are small in the CS frame [16]. Hereafter, the angles (ϑ, ϕ) and the angular coefficients A0−7 are intended to be specific to the CS rest frame. The A4 cos ϑ term violates parity, and is due to the interference of the amplitudes of the vector and axialvector currents. Its presence induces an asymmetry in the ϕ-integrated cos ϑ dependence of the cross section. Two sources contribute: the interference between the Zboson vector and axial-vector amplitudes, and the interference between the photon vector and Z-boson axialvector amplitudes. The asymmetric component from the γ∗-Z interference cross section contains gAf couplings that are independent of sin2 θW . The asymmetric component from Z-boson self-interference contains a product of gVf from the lepton and quark vertices, and thus is related to sin2 θW . At the Born level, this product is T3 (1 − 4|Q | sin2 θW ) T3q (1 − 4|Qq| sin2 θW ), where and q denote the lepton and quark, respectively. For the Drell-Yan process, the relevant quarks are predominantly the light quarks u, d, or s. The coupling factor has an enhanced sensitivity to sin2 θW at the lepton-Z vertex: for a sin2 θW value of 0.223, a 1% variation in sin2 θW changes the lepton factor (containing Q ) by about 8%, and it changes the quark factor (containing Qq) by about 1.5% (0.4%) for the u (d or s) quark. Electroweak radiative corrections do not alter significantly this Born-level interpretation. Loop and vertex electroweak radiative corrections are multiplicative form-factor corrections to the couplings that change their values by a few percent [5]. The − forward-backward asymmetry in cos ϑ is defined as σ+(M ) − σ−(M ) 3 Afb(M ) = σ+(M ) + σ−(M ) = 8 A4(M ) , (2) where M is the lepton-pair invariant mass, σ+ is the total cross section for cos ϑ ≥ 0, and σ− is the total cross section for cos ϑ < 0. Figure 2 shows the typical depen- dence of the asymmetry as a function of the lepton-pair invariant mass from a Drell-Yan QCD calculation. The offset of Afb from zero at M = MZ is related to sin2 θW . Away from the Z pole, the asymmetry is dominated by the component from γ∗-Z interference, whose cross section is proportional to (M 2 − MZ2 )/M 2, and the asymmetries in these regions are primarily related to the flux of partons. Consequently, the asymmetry distribution is sensitive to both sin2 θW and the parton distribution functions (PDF) of the proton. The sin2 θeleffpt coupling is derived from the measurement of Afb(M ) and predictions of Afb(M ) for various input values of sin2 θW . Electroweak and QCD radiative corrections are included in the predictions of Afb(M ), with the electroweak radiative corrections derived from an approach adopted at LEP [18]. 6 Afb u+ d 0.6 u 0.4 0.2 d 0 -0.2 -0.4 40 60 80 100 120 140 160 180 200 M (GeV/c2) FIG. 2. Typical dependence of Afb as a function of the lepton-pair invariant mass M . The label u + d denotes the overall asymmetry, and the labels u and d denote the contribution to the overall asymmetry from quarks with charge 2/3 and −1/3, respectively. The contribution of quarks categorized by the u or d label is (σq+ − σq−)/σ, where q = u or d, σ+(−) their forward (backward) cross section, and σ the total cross section from all quarks. The vertical line is at M = MZ . III. ENHANCED QCD PREDICTIONS Drell-Yan cross-section calculations with QCD radiation do not typically include electroweak radiative corrections. However, the QCD, quantum electrodynamic, and weak-interaction corrections can be organized to be individually gauge invariant so that they are applied as independent components. Quantum electrodynamic (QED) radiative corrections that result in final-state photons are the most important for measurements derived from the Drell-Yan process, and they are included in the physics and detector simulation described in Sec. VI. The effects of QED radiation are removed from the measured distribution of Afb using the simulation so that the measurement can be directly compared with QCD calculations of Afb that do not include it. The Drell-Yan process and the production of quark pairs in high-energy e+e− collisions are analogous processes: qq¯ → e+e− and e+e− → qq¯. At the Born level, the process amplitudes are of the same form except for the interchange of the electrons and quarks. Electroweak radiative corrections, calculated and extensively used for precision fits of LEP-1 and SLD measurements to the standard model [11, 12], are therefore applicable to the Drell-Yan process. In the remainder of this section, the technique used to incorporate independently calculated electroweak radiative corrections for e+e− collisions into existing QCD calculations for the Drell-Yan process is presented. A. Electroweak radiative corrections The effects of virtual electroweak radiative corrections are incorporated into Drell-Yan QCD calculations via form factors for fermion-pair production according to e+e− → Z → ff¯. The Z-amplitude form factors are calculated by zfitter 6.43 [18], which is used with LEP1 and SLD measurement inputs for precision tests of the standard model [11, 12]. Corrections to fermion-pair production via a virtual photon include weak-interaction W -boson loops in the photon propagator, and Z-boson propagators at fermion-photon vertices; these corrections are not gauge-invariant except when combined with their gauge counterparts in the Z amplitude. The zfitter weak and QED corrections are organized to be separately gauge-invariant. Consequently, weak corrections to fermion-pair production via the virtual photon are included through the Z-amplitude form factors. zfitter uses the on-shell scheme [3], where particle masses are on-shell, and sin2 θW = 1 − MW2 /MZ2 (3) holds to all orders of perturbation theory by definition. Since the Z-boson mass is accurately known (to ±0.0021 GeV/c2 [11, 12]), the inference of sin2 θW is equivalent to an indirect W -boson mass measurement. Form factors calculated by zfitter are tabulated for later use in QCD calculations. The specific standard- model assumptions and parameters used in the form- factor calculation are presented in the appendix, as well as their usage in the scattering amplitude Aq. The calculated form factors are ρeq, κe, κq, and κeq, where the label e denotes an electron and q denotes a quark. As the calculations use the massless-fermion approximation, the form factors only depend on the charge and weak isospin of the fermions. Consequently, the tabulated form factors are distinguished by three labels, e (electron type), u (up-quark type), and d (down-quark type). The form factors are complex valued, and are functions of the sin2 θW parameter and the Mandelstam sˆ variable of the e+e− → Z → ff¯ process. The ρeq, κe, and κq form fac- tors of the amplitude can be reformulated as corrections to the Born-level gAf and gVf couplings, gVf → √ ρeq (T3f − 2Qf κf sin2 θW ) and gAf → √ ρeq T3f , where f represents e or q. The products κf sin2 θW , called effective-mixing terms, are directly accessible from measurements of the asym- metry in the cos ϑ distribution. However, neither the sin2 θW parameter nor the sˆ-dependent form factors can be inferred from measurements without assuming the standard model. The effective-mixing terms are denoted 7 as sin2 θeff to distinguish them from the on-shell definition of the sin2 θW parameter of Eq. (3). The Drell-Yan process is most sensitive to the sin2 θeff term of the lepton vertex, κe sin2 θW . At the Z pole, κe is independent of the quark flavor, and the flavor-independent value of κe sin2 θW is commonly denoted as sin2 θeleffpt. For comparisons with other measurements, the value of sin2 θeleffpt at the Z pole is taken to be Re[κe(MZ2 )] sin2 θW . 3.0 PDFs, and are used for direct comparisons with the powheg-box calculations. The powheg-box NLO program, in conjunction with the NNPDF-3.0 NNLO PDFs, is chosen as the default EBA-based QCD calculation of Afb with various input values of sin2 θW . The resbos calculation is used as a reference for resummed calculations. The LO calculation serves as a reference calculation for the sensitivity of Afb to QCD radiation. B. QCD calculations IV. EXPERIMENTAL APPARATUS The zfitter form factors ρeq, κe, and κq are inserted into the Born gAf and gVf couplings of the Drell-Yan process. The κeq form factor is incorporated as an amplitude correction. This provides an enhanced Born approximation (EBA) to the electroweak terms of the amplitude. The form factor for the QED self-energy correction to the photon propagator is also part of the EBA. Complexvalued form factors are used in the amplitude. Only the electroweak-coupling factors in the QCD cross sections are affected. The standard LEP Z-boson resonant lineshape and the total decay width calculated by zfitter are used. Both leading-order (LO) and next-to-leading-order (NLO) QCD calculations of Afb for the process pp¯ → γ∗/Z → + − are performed with form factors provided by zfitter. Two sets of PDFs are used to provide the incoming parton flux used in all QCD calculations discussed in this section, except where specified otherwise. They are the NLO CTEQ6.6 [19] PDFs and the next-to-next-to-leading order (NNLO) NNPDF-3.0 [20] PDFs. For consistency with the zfitter calculations, the NNPDFs selected are derived with a value of the strong-interaction coupling of 0.118 at the Z mass. Two NLO calculations, resbos [21] and the powhegbox implementation [22] of the Drell-Yan process [23], are modified to be EBA-based QCD calculations. For both calculations, the cross section is finite as PT2 vanishes. The resbos calculation combines a NLO fixedorder calculation at high boson PT with the CollinsSoper-Sterman resummation formalism [24] at low boson PT, which is an all-orders summation of large terms from gluon emission calculated to next-to-next-to-leading log accuracy. The resbos calculation uses CTEQ6.6 NLO PDFs. The powheg-box calculation uses the NNLO NNPDF-3.0 PDFs, and is a fully unweighted partonic event generator that implements Drell-Yan production of + − pairs at NLO. The NLO production implements a Sudakov form factor [25] that controls the infrared diverence at low PT, and is constructed to be interfaced with parton showering to avoid double counting. The pythia 6.41 [26] parton-showering algorithm is used to produce the final hadron-level event. The combined implementation has next-to-leading log resummation accuracy. The LO calculations of Afb are based on numerical integrations of the LO cross section using NNPDF- The CDF II apparatus is a general-purpose detector [27] at the Fermilab Tevatron, a pp¯ collider with a center-of-momentum (cm) energy of 1.96 TeV. The positive-z axis of the detector coordinate system is directed along the proton direction. For particle trajectories, the polar angle θcm is relative to the proton direction and the azimuthal angle φcm is oriented about the beamline axis with π/2 being vertically upwards. The pseudorapidity of a particle is η = − ln tan(θcm/2). Detector coordinates are specified as (ηdet, φcm), where ηdet is the pseudorapidity relative to the detector center (z = 0). The momentum P of a charged particle is measured in the magnetic spectrometer, which consists of chargedparticle tracking detectors (trackers) immersed in a magnetic field. The energy of a particle is measured in the calorimeters surrounding the magnetic spectrometer. The component of momentum transverse to the beamline is PT = |P | sin θcm. The component of energy transverse to the beamline is ET = E sin θcm. The tracking detectors consist of a central tracker and an inner silicon tracker. The central tracker is a 3.1 m long, open-cell drift chamber [28] that extends radially from 0.4 to 1.4 m. Between the Tevatron beam pipe and the central tracker is a 2 m long silicon tracker [29]. Both trackers are immersed in a 1.4 T axial magnetic field produced by a superconducting solenoid just beyond the outer radius of the drift chamber. Combined, these two trackers provide efficient, high-resolution tracking and momentum measurement over |ηdet| < 1.3. Outside the solenoid is the central barrel calorimeter [30, 31] that covers the region |ηdet| < 1.1. The forward end-cap regions, 1.1 < |ηdet| < 3.5, are covered by the end-plug calorimeters [32–34]. The calorimeters are scintillator-based sampling calorimeters, which are segmented along their depth into electromagnetic (EM) and hadronic (HAD) sections, and transversely into projective towers. The EM calorimeter energy resolutions meas√ured in test beams with electrons are σ/E = 13.5%√/ ET for the central calorimeter, and σ/E = 16%/ E ⊕ 1% for the plug calorimeter, where the symbol ⊕ is a quadrature sum, and ET and E are in units of GeV. Both the central and plug EM calorimeters have preshower and shower-maximum detectors for electromagnetic-shower identification and centroid measurements. The combination of the plug shower- 8 maximum detector and silicon tracker provides enhanced tracking coverage to |ηdet| = 2.8. However, as |ηdet| increases for plug-region tracks, the transverse track-length within the magnetic field decreases, resulting in increasingly poor track-curvature resolution. Within the plug shower-maximum detector, |ηdet| = 2.8 corresponds to a radial extent from the beamline of 23 cm. V. DATA SELECTION The data set, collected over 2002–2011, is the full CDF Run II sample and consists of pp¯ collisions corresponding to an integrated luminosity of 9.4 fb−1. Section V A reports on the online selection of events (triggers) for the Afb measurement. Section V B describes the offline selection of electron candidates, and Sec. V C describes the selection of electron pairs. A. Online event selection Electron candidates are selected from two online triggers, central-18, and Z-no-track. The central-18 selection accepts events containing at least one electron candidate with ET > 18 GeV in the central calorimeter region. Candidates are required to have electromagnetic shower clusters in the central calorimeters that are geometrically matched to tracks from the central tracker. Electron candidates for the Z-no-track selection have no track requirement and are only required to be associated with an electromagnetic shower cluster with ET > 18 GeV. The selection, which accepts events containing at least one pair of candidates located in any calorimeter region, is primarily for dielectrons in the plug-calorimeter region where online tracking is not available. It also accepts the small fraction of dielectron events that fail the track requirements of the central18 trigger. B. Offline electron selection After offline event reconstruction, the purity of the sample is improved with the application of CDF standard-electron identification and quality requirements [27]. Fiducial constraints are applied to ensure that the electrons are in well-instrumented regions, thus ensuring good-quality and predictable reconstruction performance. Each electron candidate is required to be associated with a track, to significantly reduce backgrounds. The track-vertex position along the beamline (zvtx) is restricted to be within the luminous region, |zvtx| < 60 cm. Overall, 3% of the pp¯ luminous region along the beamline is outside this fiducial region. Electron identification in the central calorimeter region is optimized for electrons of PT > 10 GeV/c [27]. It uses information from the central and silicon trackers, the longitudinal and lateral (tower) segmentation of the electromagnetic and hadronic calorimeter compartments, and the shower-maximum strip detector (CES) within the electromagnetic calorimeter. The highest quality of signal selection and background rejection is provided by the trackers in combination with the CES. An electron candidate must have shower clusters within the electromagnetic calorimeter towers and CES signals compatible with the lateral profile of an electromagnetic shower. A candidate must also be associated to a track that extrapolates to the three-dimensional position of the CES shower centroid. The transverse momentum of the particle associated with the track must be consistent with the associated electron shower ET via an E/P selection when PT < 50 GeV/c [27]. For both the track matching in the CES and the E/P selection, allowances are included for bremsstrahlung energy-loss in the tracking volume, which on average is about 20% of a radiation length. The fraction of shower energy in the hadroniccalorimeter towers behind the tower cluster of the electromagnetic calorimeter must be consistent with that for electrons through an EHAD/EEM requirement. These selections are more restrictive than those applied in the online selections described in Sec. V A. Such an offline selection has high purity and is called the tight central electron (TCE) selection. To improve the selection efficiency of central-electron pairs, a looser selection, called the loose central electron (LCE) selection, is applied on the second electron candidate. The looser variant does not use transverse shower-shape constraints, the E/P constraint, or track matching in the CES. For track associations, the track is only required to project into the highest-energy calorimeter tower within the cluster of towers associated with the electromagnetic shower. Electron identification in the plug calorimeter also uses tracker information, the longitudinal and lateral (tower) segmentation of the electromagnetic and hadronic calorimeter compartments, and the showermaximum strip detector (PES) within the electromagnetic calorimeter. As the plug-calorimeter geometry differs from the central geometry, the details of the selection requirements differ. The end-plug calorimeters, with sampling planes perpendicular to the beamline, have projective towers that are physically much smaller than the central calorimeter towers and vary in size [32]. The electromagnetic showers in the plug calorimeter are clustered into “rectangular” 3 × 3 detector-tower clusters centered on the highest-energy tower. Good radial containment of these showers is achieved. The preshower detector is the first layer of the plug-electromagnetic calorimeter and it is instrumented and read out separately. As there are approximately 0.7 radiation lengths of material in front of it, its energy is always included in the electromagneticcluster shower energy. Tracks entering the plug calorimeters have limited geometrical acceptance in the central tracker for |ηdet| > 1.3. 9 The forward coverage of the silicon tracker is exploited with a dedicated calorimetry-seeded tracking algorithm called “Phoenix”. It is similar to the central trackingalgorithm, where tracks found in the central tracker are projected into the silicon tracker and hits within a narrow road of the trajectory initialize the silicon track reconstruction. With the Phoenix algorithm, the track helix in the magnetic field is specified by the position of the pp¯ collision vertex, the three-dimensional exit position of the electron into the PES, and a helix curvature. The collision vertex is reconstructed from tracks found by the trackers. The curvature is derived from the ET of the shower in the electromagnetic calorimeter. Two potential helices are formed, one for each charge. The algorithm projects each helix into the silicon tracker to initialize the track reconstruction. If both projections yield valid tracks, the higher-quality one is selected. Depending on its vertex location along the beamline, a track traverses zero to eight layers of silicon. A Phoenix track is required to traverse at least three silicon layers and have at least three silicon hits. Eighty percent of the tracks traverse four or more silicon layers, and the average track acceptance is 94%. An electron candidate in a plug calorimeter must have a shower cluster within the electromagnetic calorimeter towers, an associated PES signal compatible with the lateral profile of an electromagnetic shower, and a longitudinal profile, measured using EHAD/EEM, that is consistent with that expected for electrons. The candidate must also be associated with a Phoenix track. Neither a PT nor an E/P selection requirement is applied because the track momentum determined by the Phoenix algorithm is correlated with the calorimeter energy. Charge misidentification is significant at large |ηdet| because of the reduced track-helix curvature resolution. The resolution is inversely proportional to the track-exit radius at the PES, which varies from 23 to 129 cm. As Drell-Yan high-ET leptons are typically produced in isolation, the electron candidates are required to be isolated from other calorimetric activity. The isolation energy, Eiso, is defined as the sum of ET over towers within a 0.4 isolation cone in (η, φ) surrounding the electron cluster. The towers of the electron cluster are not included in the sum. For central-electron candidates, the isolation requirement is Eiso/ET < 0.1, and for plugelectron candidates, it is Eiso < 4 GeV. C. Offline electron-pair event selection Events are required to contain two electron candidates in either the central or plug calorimeters. These events are classified into three topologies, CC, CP, and PP, where C (P) denotes that the electron is detected in the central (plug) calorimeter. The electron kinematic variables are based on the electron energy measured in the calorimeters and on the track direction. The kinematic and fiducial regions of acceptance for electrons in the three topologies are described below. 1. Central–central (CC) (a) ET > 25 (15) GeV for electron 1 (2); (b) 0.05 < |ηdet| < 1.05. 2. Central–plug (CP) (a) ET > 20 GeV for both electrons; (b) Central electron: 0.05 < |ηdet| < 1.05; (c) Plug electron: 1.2 < |ηdet| < 2.8. 3. Plug–plug (PP) (a) ET > 25 GeV for both electrons; (b) 1.2 < |ηdet| < 2.8. The CC topology consists of TCE-LCE combinations with asymmetric ET selections on electrons 1 and 2, the electrons in the pair with the higher and lower ET, respectively. Either electron can be the TCE candidate, and its LCE partner can also be a TCE candiate because they are a subset of the LCE candidates. The asymmetric selection, an optimization from the previous measurement of electron angular-distribution coefficients [17], improves the acceptance. For the CP topology, the central electron candidate must pass the TCE selection. The PP-topology electron candidates are both required to be in the same end of the CDF II detector; Drell-Yan electrons of the PP topology on opposite ends of the CDF II detector are overwhelmed by QCD dijet backgrounds at low PT. In addition, the longitudinal separation of vertex positions of the associated tracks of the candidates is required to be within 4 cm of each other. The measurement of Afb is based on the direction of the e− in the CS frame, and any charge misidentification dilutes the result. Charge misidentification is small for central tracks and significant for plug tracks. Consequently, only CC- and CP-topology pairs are used in the measurement. For the CP-topology, the central electron is used to identify the e−. Electron pairs of the PP topology are only used for plug-calorimeter calibrations and cross checks. The same-charge pairs of the CC topology are also not used in the measurement, but they are used for calibrations, simulation tuning, and consistency checks. Signal events intrinsically have no imbalance in the total energy in the transverse plane from undetected particles except for those within uninstrumented regions of the detector or from semileptonic decays of hadrons. The transverse energy imbalance E/T is the magnitude of − i ETi nˆi, where the sum is over calorimeter towers, nˆi is the unit vector in the azimuthal plane that points from the pp¯ collision vertex to the center of the calorimeter tower i, and ETi is the corresponding transverse energy in that tower. Events with E/T < 40 GeV are selected, therefore poorly reconstructed signal events, characterized by large E/T, are removed. Only a very 10 small fraction of signal events is removed. About half of the background events containing leptonically decaying W -bosons are removed because they have large intrinsic E/T from neutrinos, which are undetected. VI. SIGNAL SIMULATION both the data and simulation, and modeling and removal of backgrounds from the data. The detector has 1440 EM calorimeter towers, each with different responses over time and position within the tower. Many instrumental effects are correlated, and the overall correction and calibration process is iterative. Drell-Yan pair production is simulated using the pythia [35] Monte Carlo event generator and CDF II detector-simulation programs. pythia generates the hard, leading-order QCD interaction qq¯ → γ∗/Z, simulates initial-state QCD radiation via its parton-shower algorithms, and generates the decay γ∗/Z → + −. The CTEQ5L [36] PDFs are used in the calculations. The underlying-event and boson-PT parameters are derived from the pythia configuration pytune 101 (AW), which is a tuning to previous CDF data [35, 37, 38]. Generated events are first processed by the event simulation, and then followed by the CDF II detector simulation based on geant-3 and gflash [39]. The event simulation includes photos 2.0 [40, 41], which adds final-state QED radiation (FSR) to decay vertices with charged particles (e.g., γ∗/Z → ee). The default implementation of pythia plus photos (pythia+photos) QED radiation in the simulation has been validated in a previous 2.1 fb−1 measurement of sin2 θeleffpt using Drell-Yan electron pairs [5]. The pythia+photos calculation is adjusted using the data and the resbos calculation. The generator-level PT distribution of the boson is adjusted so that the shape of the reconstruction-level, simulated PT distribution matches the data in two rapidity bins: 0 < |y| < 0.8 and |y| ≥ 0.8. For this adjustment, reconstructed ee pairs of all topologies (CC, CP, and PP) in the 66–116 GeV/c2 mass region are used. The generator-level boson-mass distribution is adjusted with a mass-dependent K-factor. The K-factor is the ratio of the resbos boson-mass distribution calculated using CTEQ6.6 PDFs relative to the pythia 6.4 [26] boson-mass distribution calculated using CTEQ5L PDFs. No kinematic restrictions are applied. Standard time-dependent beam and detector conditions are incorporated into the simulation, including the p and p¯ beamline parameters; the luminous region profile; the instantaneous and integrated luminosities per data-taking period; and detector component calibrations, which include channel gains and malfunctions. The simulated events are reconstructed, selected, and analyzed in the same way as the experimental data. VII. DATA AND SIMULATION CORRECTIONS In this section, time- and position-dependent corrections and calibrations to the experimental and simulated data are presented. They include event-rate normalizations of the simulation to the data, energy calibrations of A. Event rate normalizations The simulation does not model the trigger and reconstruction efficiences observed in the data with sufficient precision. Time-, detector-location-, and luminositydependent differences are observed. To correct the observed differences in rate between the data and simulation, a scale-factor event weight is applied to simulated events. The scale factor is the ratio of the measured offline-selection efficiencies observed in data to the simulation versus time, detector location, and instantaneous luminosity. The base correction described above using measured efficiencies is inadequate for the Afb measurement for two reasons: 1) due to the more stringent selection requirements for the efficiency measurements, the bin sizes for the time, position, and luminosity dependence are wide, and a finer resolution is needed; and 2) the Tevatron pp¯ luminosity profile is difficult to simulate. The secondlevel correction uses event-count ratios between the data and simulation, or scale factors, as event weights. Events are required to pass all standard selection requirements and the ee-pair mass is required to be within the 66–116 GeV/c2 range. Events are separated into the CC, CP, and PP topologies and corrected separately. The time and luminosity dependencies are related. The distributions of the number of pp¯ collision vertices in each event (nvtx) and the location of these vertices along the beamline (zvtx) changed significantly with improvements to the Tevatron collider. These distributions are inadequately simulated and are corrected separately. For the nvtx correction, the data and simulation are grouped into 39 calibration periods, and the distribution corrected on a period-by-period basis. The correction of the zvtx distribution is organized into a smaller set of seven time intervals corresponding to improvements in the Tevatron collider. The zvtx distribution has an rms spread of 30 cm, and it needs to be simulated accurately because at large |zvtx| the electron acceptance as a function of ET changes significantly. The second-level correction to remove detectorlocation dependencies is a function of |ηdet|. In the central calorimeter, the corrections are small. In the plug calorimeters, the corrections are larger and they correct the effects of tower-response differences between data and simulation near tower boundaries. 11 Events / 0.25 GeV/c2 7000 6000 5000 4000 3000 2000 1000 0 70 75 80 85 90 95 100 105 110 115 M (GeV/c2) FIG. 3. Invariant ee-mass distribution for opposite-charged CC events prior to the calibration and background subtractions. The crosses are the data and the solid histogram is the simulation. Events / 0.25 GeV/c2 8000 7000 6000 5000 4000 3000 2000 1000 0 70 75 80 85 90 95 100 105 110 115 M (GeV/c2) FIG. 4. Invariant ee-mass distribution for CP events prior to the calibration and background subtractions. The crosses are the data and the solid histogram is the simulation. B. Energy calibrations The energy calibrations are relative to the standard calibrations for time-dependent beam and detector conditions. Energy calibrations are multidimensional, and since it is not feasible to calibrate all components simultaneously, they are iteratively calibrated with a sequence of four steps using groups of lower dimension. The standard calibrations for the calorimeter have energy-scale miscalibrations that depend on time and detector location, and range up to 5% in magnitude. The miscalibrations differ for the data and the simulation, and are larger at the edges of the plug calorimeter. The energy resolution of the simulation also needs additional tuning. Without any adjustments, the mass distributions of CC- and CP-topology electron pairs are as shown in Figs. 3 and 4, respectively. Adjustments to correct the miscalibrations are con- strained using the mass distributions of e+e− pairs about the Z pole. Calibration adjustments are based on three electron-pair mass distributions: 1) generator level, 2) simulated data, and 3) data. All three mass distributions are fit to a line shape that includes the Z-pole mass as a fit parameter. The Z-pole mass values obtained from fits to the experimental and simulated data are separately aligned to the corresponding generator-level value [42]. The generator-level mass is evaluated using clustered energies and includes the effects of QED FSR. The FSR electrons and photons are clustered about the seed tower in a manner similar to the clustering of electron reconstruction. The seed tower is based on the reconstructed electron, and the projection from the pp¯ collision vertex to the tower is achieved by extrapolating the track helix. Since the detector acceptance slightly alters the lineshape of the mass distribution, generator-level events are selected by requiring that their kinematic properties after detector simulation meet all selection criteria. The generator-level mass distribution is fit to the standard LEP Z-boson resonant line shape. The data and simulation mass distributions are fit to the standard LEP Z-boson resonant line shape convoluted with the resolution functions of the calorimeters, which are Gaussian. Typically, the fit range is ±5 GeV/c2 around the Z peak. The Z-pole mass and resolution width values are allowed to vary but the resonant width is fixed to the corresponding generator-level fit value. With this method, the resolution width values of the simulation and data are directly comparable and are used to calibrate the energy resolution of the simulation to the data. Electron pairs of all topologies that satisfy the selection requirements are used in the calibration. The set of CC+PP events, and separately, of CP events, provides two independent sets of calibrations for all calorimeter components, such as towers. The electrons used to calibrate the energy scale of a component are denoted as reference electrons. The partners to these electrons can be anywhere in the detector so that miscalibrations of the current iteration are averaged out, and they also serve as references elsewhere. Energy-scale adjustments require the constraint of the sharp, nearly symmetric peak at the Z pole of the mass distribution. The energy distribution of the electrons is not as suitable because it is broad and asymmetric, and sensitive to the boson transverse-momentum and rapidity distributions, as well as the e+e− angular distribution. The first step in an iteration is the time-dependent calibration of the overall energy scales for the central and plug calorimeters. Corrections are determined for each of the 39 calibration periods introduced in the previous section. The next step is the relative calibration of calorimeter towers and the response maps within their boundaries. In this step, the bins are small and do not have enough events for accurate mass fits. Consequently, the energy response for each bin is quantified using the statistically more accurate mean of the scaled electron-pair 12 mass M/(91.15 GeV/c2) over the range 0.9–1.1, and is normalized to the overall average of the central or plug calorimeters. Tower-response corrections are important in the high |ηdet| region of the plug calorimeters where standard calibrations are difficult. Corrections are determined for two time periods: calibration periods 0–17 and 18–39. Period 18 is the start of consistently efficient high-luminosity Tevatron operations, which commenced from April 2008. Both the central and plug calorimeter towers require additional response-map tuning at the periphery of the towers. The third step calibrates the energy scales of the ηtower rings of the calorimeter. A ring consists of all towers in the φ dimension with the same |ηdet| dimension. The adjustments from this step isolate the systematic variation of the energy scale in the η dimension of the standard calibration relative to the underlying physics. There are 22 η-tower rings, 12 of which are in the plug calorimeter. The lowest and highest η-tower rings of the plug calorimeter are not in the acceptance region. Separate calibrations for the CC+PP and CP data are iteratively determined using two passes, with corrections determined for two time periods, 0–17 and 18–39. First, the central and plug rings are calibrated with events from the CC+PP data. These calibrations are used only for CC- and PP-topology pairs. Then the CP data set calibration is derived from CP events, using the CC+PP calibrations as initial values for the calibration. The calibrations from the CC+PP and CP sets are expected to be slightly different due to the wide zvtx distribution of pp¯ collision vertices at the Tevatron. The geometry of an electron shower within the CDF calorimeters depends on the position of the collision vertex. Away from zvtx = 0, the transverse segmentation of the calorimeter is less projective, and the fraction of the shower energy sampled by the calorimeter is different. As the magnitude of zvtx increases, the electron energy reconstructed in the calorimeter can change. Accompanying the η-ring correction is the extraction of the underlying-event energy contained within an electron-shower cluster. The electron-pair mass distributions show an observable dependence on the number of pp¯ collision vertices in an event. Assuming that the underlying-event energy per shower cluster increases linearly with nvtx, these mass distributions are used to extract the associated underlying-event energy of a shower cluster per vertex for each η ring. For the central calorimeter, the value is approximately constant at 35 MeV. For the plug calorimeters, the value is approximately 150 MeV for |ηdet| < 2 and increases to 1.5 GeV at |ηdet| ≈ 2.8. The expected underlying-event energy is subtracted from the measured electron energy. The fourth step removes residual miscalibrations in both ηdet and φ. The energy scales on a grid with 16 ηdet and 8 φcm bins are calibrated, along with determinations of the corresponding energy resolutions. The ηdet bins span both ends of the detector, with eight bins each for the central and plug calorimeters. Events in each Energy correction 1.03 1.02 1.01 1 0.99 0.98 0.970 5 10 15 20 25 30 35 Calibration period FIG. 5. Corrections to the global energy scales as functions of the calibration period for the data. The central calorimeter corrections are the crosses (blue), and for the plug calorimeter, the histogram (red) gives the corrections. (ηdet, φcm) bin are further divided into electron pairs with ηdet values of the same sign (SS) and pairs with oppositesign values (OS). There are differences of a few tenths of a percent between the SS- and OS-pair calibrations. The electrons of SS and OS pairs also have differing showering geometries within the calorimeters due to the wide zvtx distribution of pp¯ collisions. The fraction of SS pairs for the CC topology is approximately 50%. For the CP topology, the fraction varies with the η-bin index, and the range is approximately 50% to 80%. The PP-topology sample consists entirely of SS pairs. The energy resolution of the calorimeter simulation is also adjusted for each calibration bin of the fourth step. Line-shape fits to the mass distributions of the data and the simulation provide the resolution-smearing parameters σd2 and σs2, respectively. The fit values of σd are approximately 2 GeV/c2 for all bins. For most bins, the simulation resolution is adjusted with an additional Gaussian rms deviation of σd2 − σs2. For 24% of the central bins, this value is negative, and the alternative is to rescale the simulation energy bias ∆Ebias ≡ Egen − Erec of each event, where Egen is the generator-level clustered energy and Erec is the reconstruction-level energy. The resolution is modified by scaling the event-by-event bias with the factor fbias so that the new reconstruction-level energy is Egen − fbias∆Ebias. The value of fbias does not deviate from its expected value of unity by more than 17%. The energy calibration stabilizes after three iterations. The time-dependent global corrections to the energy scales of the central and plug calorimeters from step one are shown in Fig. 5. Approximately 20% of the data is contained in time periods 0–10, and 68% in time periods 18-38. The energy calibrations over η-tower rings from step three have the largest effect. Figure 6 shows the corrections derived from the CP calibration set for the two time periods, 0–17 and 18–38. The corrections de- 13 Energy correction 1.06 1.05 1.04 1.03 1.02 1.01 1 0.99 0.980 2 4 6 8 10 12 14 16 18 20 22 Index: η-tower ring FIG. 6. Corrections to the energy scales as functions of the calorimeter η-tower ring index for the CP-topology data. The corrections from time periods 0 to 17 are represented by the histograms (blue), and those from time periods 18 to 38 by the crosses (red). The central calorimeter region is index 0 to 9. Index 21 towers are about 23 cm from the beamline. rived from the CC+PP calibration set are similar. For the central calorimeter, the corrections from periods 0 to 17 and 18 to 38 are different because its standard calibration procedure was modified prior to the start of period 18. The tower-gain calibrations include an additional ηdependent correction that ranges from 0 to 2%. For periods 0–10 and 11–17, the central-calorimeter corrections are close to and compatible with the combined corrections shown in Fig. 6. The mass distributions of CCand CP-topology electron pairs after the energy calibration adjustments, and other corrections presented next, are shown later in Figs. 12 and 13, respectively. C. Backgrounds The backgrounds are negligible in the Z-peak region used for the energy calibration but they are detectable in the low- and high-mass regions of the mass distributions. In this section, the level and shapes of the backgrounds in the ee-pair mass distribution are determined separately for each of the CC, CP, and PP topologies. The backgrounds are from the production of QCD dijets, Z → τ +τ −, W +jets, dibosons (WW, WZ, ZZ ), and tt¯ pairs. All backgrounds except for QCD are derived from pythia [35] samples that are processed with the detector simulation, and in which the integrated luminosity of each sample is normalized to that of the data. The diboson and tt¯ sample normalizations use total cross sections calculated at NLO [43]. The W +jets and Z → τ +τ − sample normalizations use the total cross sections calculated at LO multiplied by an NLO-to-LO K-factor of 1.4. Sample normalizations based on these calculated cross sections are referenced as default normalizations. Simulated events are required to pass all selections required of the data. The QCD background is primarily from dijets that are misidentified as electrons. This background is extracted from the data assuming that its combination with the sum of the simulated signal and other backgrounds matches the observed mass distribution. The QCD background distribution, parametrized with level and shape parameters, is determined in a fit of the data to the sum of all backgrounds in conjunction with the simulated signal. The mass range for the fit is 42–400 GeV/c2 with 50 equally spaced bins in ln M , and the minimization statistic is the χ2 between the data and the sum of predicted components over all bins. The normalizations of the simulated signal and backgrounds are also allowed to vary from their default values via scale factors. However, as most simulated backgrounds are very small, they are only allowed to vary within their normalization uncertainties. The constraint is implemented with an additional χ2 term (fnorm − 1)2/0.0852, where fnorm is the scale factor of the background calculation. The uncertainty of the measured luminosity is 6% [44]; the prediction uncertainty is taken to be equally as large; and their combination gives the estimate for the constraint uncertainty of 0.085. The tt¯, diboson, and W +jets backgrounds are always constrained. The Z → τ +τ − background is the second largest, and for CC-topology events, the scale factor is determined with the data. However, for CP- and PP-topology events, the Z → τ +τ − background scale factors are constrained to their default normalizations. For the QCD-background analysis, two independent data samples are used: events passing the selection criteria and events failing them. The first sample, denoted as the signal sample, is used to determine the level of the QCD background and its shape over the mass distribution. The second sample, denoted as the QCDbackground sample, is derived from events failing the selection criteria, and is for the event-by-event background subtractions from kinematic distributions. Electron-like candidates for the QCD-background samples are selected by reversing criteria that suppress hadrons and QCD jets. One candidate passes all electron selection requirements except the isolation criterion. The other is required to be “jet-like” by reversing the isolation and EHAD/EEM selection criteria. Since there is a small fraction of γ∗/Z events in the initial background sample, the reverse selections are optimized for each eepair topology to remove them. As the reversed selection criteria bias the mass distributions, events of these QCDbackground samples are reweighted so that the overall normalization and the shapes of the mass distributions match those extracted from the signal samples. For the CC topology, same-charge pairs passing the selection criteria are also used to determine the QCD background parameters, because 50%-60% of the events in the low- and high-mass regions are from QCD. The first step in the background determination is the extraction of the shape and default level of the QCD background 14 Events / 0.045 Events / 0.045 105 104 103 102 10 1 10-1 10-2 3.8 4 4.2 4.4 4.6 4.8 5 5.2 5.4 5.6 5.8 ln [M (GeV/c2)] 105 104 103 102 10 1 10-1 10-2 3.8 4 4.2 4.4 4.6 4.8 5 5.2 5.4 5.6 5.8 ln [M (GeV/c2)] FIG. 7. Logarithmically binned mass distributions for oppo- sitely charged ee-pair candiates in CC-topology events. The (black) crosses are the data, the (red) histogram overlapping the data is the sum of all components, the (green) histogram concentrated at lower masses is the Z → τ +τ − component, and the (cyan) histogram in the middle with the Z peak is the diboson component. The remaining broad distributions, from top to bottom, are QCD (magenta), W +jets (blue), and tt¯ (purple). The comparison of the data with the sum of the components yields a χ2 of 56 for 50 bins. FIG. 8. Logarithmically binned mass distributions for CP- topology electron-pair candidates. The (black) crosses are the data, the (red) histogram overlapping the data is the sum of all components, the (green) histogram concentrated at lower masses is the Z → τ +τ − component, and the (cyan) his- togram in the middle with the Z peak is the diboson compo- nent. The remaining broad distributions, from top to bottom are: QCD (magenta), W +jets (blue), and tt¯ (purple). The comparison of the data with the sum of the components yields a χ2 of 50 for 50 bins. from the same-charge distribution. Then, the mass distributions of both same-charge and opposite-charge pairs passing the selection criteria are fit simultaneously for the background level parameters. The large fraction of QCD events in the same-charge distribution constrains the QCD background level parameter. Consequently, the scale factor for the normalization of the Z → τ +τ − background is determined using the data, but the default normalization is not accommodated. If the Z → τ +τ − normalization is allowed to vary, the fit determines a scale factor value of 0.53 ± 0.11. However, if the normalization is restricted via the constraint to the default value, the fit pulls the scale factor away from its default value of unity to a value of 0.83±0.07, and the χ2 increases by six units relative to the unconstrained fit. The detector simulation and event normalizations for the Z → τ +τ − sample, consisting of lower-ET secondary electrons from τ decays, are not tuned. Consequently, the 0.53 value is chosen for the Afb measurement and the 0.83 value is used as a systematic variation. The CC-topology opposite-charge mass distributions for the data, the simulated data plus backgrounds, and the backgrounds are shown in Fig. 7. For the CP and PP topologies, the signal samples consist of both same- and opposite-charge electron pairs. Charge separation is not useful because of the significant charge misidentification rate for electrons in the plug region. The largest background in each topology is from QCD. However, the sum of all backgrounds is still small in relation to the signal. If all backgrounds are allowed to vary in the fits, the minimizations are underconstrained. Consequently, the simulated backgrounds are constrained to their default normalizations, and only the levels and shapes of the QCD backgrounds are varied. The shape of the QCD background for each topology is parametrized with an asymmetric-Gaussian function that consists of two piecewise continuous Gaussians joined at their common mean but with different widths. One of the function parameters is empirically tuned in the high- or low-mass region. As these regions have the largest level of backgrounds, it is important to control the fit within these regions. For the CP topology, the width on the high-mass side is first optimized in the region M > 127 GeV/c2, and then the backgrounds and simulated signal are fit to the data. For the PP topology, the mean of the asymmetricGaussian is first optimized in the low-mass region in the vicinity of the mass threshold, and then the backgrounds and simulated signal are fit to the data. The CP- and PP-topology mass distributions for the data, the simulated data plus backgrounds, and the backgrounds are shown in Figs. 8 and 9, respectively. The CC-, CP-, and PP-topology samples contain approximately 227 000, 258 000, and 80 000 events, respectively, within the 42–400 GeV/c2 mass region. Table I summarizes the overall background levels for these samples. The total backgrounds for CC-, CP-, and PPtopology samples are 1.1%, 1.2%, and 2.1%, respectively. For the CC- and CP-topology samples shown in Figs. 7 and 8 respectively, the background fractions in the vicinity of the Z-pole mass are small, but away from the pole mass, the levels are larger and range from about 0.1% to about 10%. 15 Events / 0.045 104 103 102 10 1 10-1 10-2 3.8 4 4.2 4.4 4.6 4.8 5 5.2 5.4 5.6 5.8 ln [M (GeV/c2)] FIG. 9. Logarithmically binned mass distributions for PPtopology ee-pair candidates. The (black) crosses are the data, the (red) histogram overlapping the data is the sum of all components, the (green) histogram concentrated at lower masses is the Z → τ +τ − component, and the (cyan) histogram in the middle with the Z-peak is the diboson component. The remaining broad distributions, from top to bottom are: QCD (magenta), W +jets (blue), and tt¯ (purple). The comparison of the data with the sum of the components yields a χ2 of 69 for 50 bins. where ∆ is the difference between the − and + momentum vectors; RT is the transverse unit vector along Pp × P , with Pp being the proton momentum vector and P the lepton-pair momentum vector; and PT is the unit vector along the transverse component of the lepton-pair momentum vector. At PT = 0, the angular distribution is azimuthally symmetric. The defintions of cos ϑ and tan ϕ are invariant under Lorentz boosts along the laboratory z direction. Afb is measured in 15 mass bins distributed over the range 50 < M < 350 GeV/c2. This section details the measurement method and presents the fully corrected measurement. Section VIII A describes a newly developed event-weighting technique. Section VIII B describes final calibrations and presents comparisons of the data and simulation. Section VIII C describes the resolutionunfolding technique and the corresponding covariance matrix of the unfolded Afb measurement. Section VIII D describes the final corrections to the measurement and presents the fully corrected measurement of Afb. A. Event-weighting method TABLE I. Background fractions within the 42–400 GeV/c2 mass region. The values with uncertainties are derived from the data. Component QCD Z → ττ W +jets Diboson tt¯ Background fraction (%) CC CP PP 0.55±0.03 0.69 ± 0.13 1.64 ± 0.28 0.26±0.06 0.21 0.27 0.13 0.16 0.10 0.14 0.10 0.08 0.02 0.01 0.01 The forward-backward asymmetry Afb of Eq. (2) is typically determined in terms of the measured cross section σ = N/(L A), where N is the number of observed events after background subtraction, L the integrated luminosity, the reconstruction efficiency, and A the acceptance within the kinematic and fiducial restrictions. The expression is N +/( A)+ − N −/( A)− Afb = N +/( A)+ + N −/( A)− , VIII. THE Afb MEASUREMENT The Collins-Soper frame angle, cos ϑ [15], is reconstructed using the following laboratory-frame quantities: the lepton energies, the lepton momenta along the beam line, the dilepton invariant mass, and the dilepton transverse momentum. The angle of the negatively charged lepton is cos ϑ = l+−l−+ − l−−l++ , M M 2 + PT2 where l± = (E ± Pz) and the + (−) superscript specifies that l± is for the positively (negatively) charged lepton. Similarly, the Collins-Soper expression for ϕ in terms of laboratory-frame quantities is tan ϕ = M 2 + PT2 ∆ · RT , M ∆ · PT where the terms N +(−) and ( A)+(−) respectively represent N and A for e+e− pairs with cos ϑ ≥ 0 (cos ϑ < 0), and the common integrated luminosity is factored out. Systematic uncertainties common to ( A)+ and ( A)− cancel out. The asymmetry in this analysis is measured using the event-weighting method [45], which is equivalent to measurements of Afb in | cos ϑ| bins with these simplifying assumptions: ( A)+ = ( A)− in each | cos ϑ| bin, and Eq. (1) describes the angular distributions. As the interchange of the charge labels of the electrons transforms cos ϑ to −cos ϑ, the detector assumption is equivalent to the postulate of a charge-symmetric detector for single electrons. For high PT electrons with the same momenta, regions of the detector with charge-asymmetric acceptances and efficiencies are small. Thus, to first order, the acceptance and efficiency cancel out with the event-weighting method, and the small portions that do not cancel out are later corrected with the simulation (Sec. VIII D). The measurement of Afb within a | cos ϑ| bin only de- 16 pends on the event counts N ± and is N+ − N− 8 | cos ϑ| Afb = N + + N − = 3 Afb 1 + cos2 ϑ + · · · , (4) where 1+cos2 ϑ+· · · denotes symmetric terms in Eq. (1). The event difference is proportional to 2A4| cos ϑ|, and the event sum to 2(1 + cos2 ϑ + · · · ). Each bin is an inde- pendent measurement of 8 3 Afb, or equivalently, A4, with an uncertainty of σ /ξ, where σ is the statistical uncer- tainty for Afb, and ξ the angular factor in the parentheses of Eq. (4). When the measurements are combined, the statistical weight of each bin is proportional to ξ2. The binned measurements are reformulated into an un- binned, event-by-event weighted expression Afb = Nn+ Nd+ − Nn− + Nd− . (5) The Nn± and Nd± terms represent weighted event counts, and the subscripts n and d signify the numerator and denominator sums, respectively, which contain the same events but with different event weights. Consider the N + and N − events of the binned measurement of Afb with a specific value of | cos ϑ|. In the unbinned measure- ment, their numerator and denominator weights contain: 1) factors to cancel the angular dependencies of their event difference (N + −N −) and sum (N + +N −), respec- tively, and 2) the ξ2 factor for the statistical combination of these events with events from other angular regions. The method is equivalent to using a maximum-likelihood technique, and for an ideal detector the statistical preci- sion of Afb is expected to be about 20% better relative to the direct counting method [45]. However, detector resolution and limited acceptance degrade the ideal gain. While the discussion of events weights illustrates an asymmetry measurement, the event weights presented in Ref. [45] and used in this analysis are for the measure- ment of the A4 angular coefficient. The numerator and denominator event weights for the measurement of A4 are 0.5 | cos ϑ|/ω2 and 0.5 cos2 ϑ/ω3, respectively, where ω is the symmetric 1 + cos2 ϑ + · · · term of Eq. 4. The event weights are functions of the reconstructed kinematic variables cos ϑ, ϕ, and the lepton-pair variables M and PT. Only the A0 and A2 terms of Eq. (1) are used in the denominator of the angular factor of Eq. (4), and the angular coefficients are parametrized with A0 = A2 = kPT2 kPT2 + M 2 , where k is a tuning factor for the PT dependence of the A0 and A2 coefficients. For this analysis, k = 1.65, which is derived from a previous measurement of angular coef- ficients [17]. The exact form of these angular terms in the event weights is not critical for Afb because the bulk of the events is at low boson PT. The background events are subtracted from the weighted event sums on an event-by-event basis by as- signing negative event weights when combining with the event sums. The event-weighting method also does not compensate for the smearing of kinematic variables due to the detector resolution, and the restricted sampling of the asymmetry in kinematic regions with limited acceptance. Resolution-smearing effects are unfolded with the aid of the simulation, and sampling limitations are separately compensated. B. Final calibrations Relative to the expected asymmetry distribution illustrated in Fig. 2, the observed distribution is diluted by the detector resolution and QED FSR. The dilution from the detector resolution is visible in the vicinity of the Zboson pole mass. The dilution from QED FSR is more pronounced at low masses because the rate of events produced in the vicinity of the Z-boson pole mass that radiate and are reconstructed in this low-mass region is more significant in relation to the intrinsic production rate. Detector miscalibrations add further distortions. All sources directly affect the electron-pair mass distributions that are primary inputs to the Afb(M ) distribution. The precision calibrations of both the data and simulation remove the additional distortions. In conjunction, the data-driven adjustments to the simulation remove differences between the data and simulation that impact the fully corrected Afb(M ) measurement. The Collins-Soper cos ϑ distribution for the simulation is also adjusted to improve agreement with the data. Only the symmetric part of the distribution is adjusted. The adjustments, determined for six electron-pair invariant mass bins whose boundaries are aligned with those used in the measurement, are determined from the ratios of the data-to-simulation cos ϑ distributions. The ratios are projected onto the first five Legendre polynomials: Σii==40 piPi(cos ϑ), where pi are projection coefficients and Pi(cos ϑ) are Legendre polynomials. The ratios are normalized so that the event count in the mass bin matches that of the data. The symmetric parts of the projections describe the ratios well and are used as the adjustments. Separate adjustments are applied to the CC- and CPtopology electron pairs as event weights. The corrections are a few percent or smaller in regions where the acceptance is large. Figure 10 shows the cos ϑ distributions after all calibrations for the combination of the CC and CP topologies and for the CC topology alone. The CP-topology dielectrons are dominant at large | cos ϑ| and significantly reduce the statistical uncertainty of the measurement. Figure 11 shows the Collins-Soper ϕ distribution. The CC- and CP-topology electron-pair mass distributions in the range of 66–116 GeV/c2 are shown in Figs. 12 and 13, respectively. For PP-topology electron pairs with masses in the same range, the comparison of the simulation with the data yields a χ2 of 232 for 200 bins. The electron ET distributions of the data are reasonably well described by the simulation. Figure 14 shows 17 Events / 0.02 7000 6000 5000 4000 3000 2000 1000 0-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 cos ϑ Events / 0.25 GeV/c2 7000 6000 5000 4000 3000 2000 1000 0 70 75 80 85 90 95 100 105 110 115 M (GeV/c2) Events / 0.0628 FIG. 10. Distributions of cos ϑ in the Collins-Soper frame for dielectrons with 66 < M < 116 GeV/c2. The crosses are the background-subtracted data and the histograms are simulated data. The upper pair of crosses and histogram is from the combination of the CC and CP topologies, and the lower pair is the contribution from the CC topology only. 6000 5000 4000 3000 2000 1000 0 -3 -2 -1 0 1 2 3 ϕ FIG. 11. Distibution of ϕ in the Collins-Soper frame for CCand CP-topology dielectrons with 66 < M < 116 GeV/c2. The crosses are the background-subtracted data and the solid histogram is the simulation. FIG. 12. Invariant ee-mass distribution for opposite-charged CC events. The crosses are the background-subtracted data and the solid histogram is the simulation. The comparison of the simulation with the data yields a χ2 of 214 for 200 bins. Events / 0.25 GeV/c2 8000 7000 6000 5000 4000 3000 2000 1000 0 70 75 80 85 90 95 100 105 110 115 M (GeV/c2) FIG. 13. Invariant ee-mass distribution for CP events. The crosses are the background-subtracted data and the solid his- togram is the simulation. The comparison of the simulation with the data yields a χ2 of 235 for 200 bins. 10000 Events / 0.5 GeV the ET distribution of the electron with the higher ET for CC-topology dielectrons for both the data and the simulation. Figure 15 shows the equivalent distribution for CP-topology electrons; here the electron can either be the central or plug electron. The mass distribution of CC same-charge dielectrons has a clear Z-boson peak from charge misidentification. Figure 16 shows the CC same-charge mass distribution of the data and the simulation. This figure confirms that charge misidentification is reproduced well by the detector simulation. The misidentification rate per central electron is 0.6%. Charge misidentification on the central electron of CP pairs is thus expected to be small and properly simulated. Charge misidentification, other categories of event mis- 8000 6000 4000 2000 0 30 40 50 60 70 80 90 100 ET (GeV) FIG. 14. ET distribution for the CC-topology electron with the larger ET. The crosses are the background-subtracted data and the solid histogram is the simulation. 18 Events / GeV/c2 Events / 0.5 GeV 6000 5000 4000 3000 2000 1000 020 30 40 50 60 70 80 90 100 ET (GeV) FIG. 15. ET distribution for the CP-topology electron with the larger ET. The crosses are the background-subtracted data and the solid histogram is the simulation. 300 250 200 150 100 50 0 40 50 60 70 80 90 100 110 120 130 140 M (GeV/c2) FIG. 16. Invariant ee-mass distribution for same-charge CC events. The crosses are the background-subtracted data and the solid histogram is the simulation. reconstruction, and detector resolution affect the observed value of cos ϑ. The bias of the observed value, ∆ cos ϑ, obtained from the simulation, is defined as the difference between its true value prior to the application of QED FSR and the observed value. The measurement of Afb is in turn biased by the fraction of events for which the sign of the observed cos ϑ differs from the true value; this change of sign is denoted by sign-reversed cos ϑ. The bias distribution consists of a narrow central core of well reconstructed events, and a very broad distribution from events where the electron kinematic properties are poorly reconstructed. Charge misidentification reverses the sign of cos ϑ. If events with charge misidentification are excluded, the bias distributions of CC- and CP-topology events have narrow non-Gaussian central cores centered at zero with 95% of the events being contained within the range |∆ cos ϑ| < 0.006. For oppositecharge CC-topology events, the effects of detector resolution dominate the bias. The fraction of events with Events / 0.02 4500 4000 3500 3000 2500 2000 1500 1000 500 0 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 y FIG. 17. Rapidity distribution of electron pairs from the CC and CP topologies with 66 < M < 116 GeV/c2. The crosses are the background-subtracted data and the histogram is the simulation. The upper curve is the (arbitrarily normalized) shape of the underlying rapidity distribution from pythia. The measurement of Afb is restricted to be within the region |y| < 1.7. sign-reversed cos ϑ is 0.3%, with most of the events being within the range | cos ϑ| < 0.1. For CP-topology events, the misidentification of the central-electron charge dominates the fraction of events with sign-reversed cos ϑ. The fraction decreases in value from 0.6% to 0.2% as | cos ϑ| increases from 0.2 to 0.8. The measurement resolution of cos ϑ consists of multiple components but their effects are small. The rapidity distribution of electron pairs for the asymmetry measurement is shown in Fig. 17, along with the shape the underlying rapidity distribution from pythia. At large values of |y|, the detector acceptance is significantly reduced. For increasing values of |y| in the |y| 1 region, the asymmetry slowly changes. This change can only be tracked by event-weighting method if it has the events to do so. Consequently, the measurement of Afb is restricted to the kinematic region |y| < 1.7. QCD calculations of Afb used for comparisons with the measurement are similarly restricted. The electron-pair mass range of the measurement, 50 to 350 GeV/c2, corresponds to maximum |y| values of 3.7 to 1.7, respectively. C. Resolution unfolding After applying the calibrations and corrections to the experimental and simulated data, the asymmetry is measured in 15 bins of the electron-pair invariant mass. The bin boundaries are 50, 64, 74, 80, 84, 86, 88, 90, 92, 94, 96, 100, 108, 120, 150, and 350 GeV/c2. The 50–64 and 150–350 GeV/c2 bins are referenced in plots as the underflow and overflow bins, respectively, because they include candidates reconstructed with masses outside the range of the plot. This measurement, denoted as raw because 19 0.6 0.4 Raw Afb 0.2 0 -0.2 First/last bin: underflow/overflow Data Simulation: PYTHIA+PHOTOS Prediction: PYTHIA |y| < 1.7 -0.4 60 70 80 90 100 110 120 130 140 150 M (GeV/c2) FIG. 18. Raw Afb measurement in bins of the electronpair invariant mass. The vertical line is at M = MZ . Only statistical uncertainties are shown. The pythia prediction for |y| < 1.7 does not include the effects of QED FSR. the effects of the detector resolution and final-state QED radiation are not removed, is shown in Fig. 18. As the mass resolution smearing of the calorimeter in the vicinity of the Z-boson mass has an rms of approximately 2 GeV/c2, the calibrations and tuning of the data and simulation are important for the resolution unfolding. The CC and CP events have different geometries and resolutions so they are kept separate in the eventweighting phase and the unfolding phase. They are combined for the Afb measurement and calculation of the measurement covariance matrix. The unfolding of the resolution and QED FSR uses the event transfer matrices from the simulation, denoted by n¯gr. All data-driven corrections to the simulation are included. The symbol n¯gr identifies the number of selected events that are generated in the electron-pair (M, cos ϑ) bin g and reconstructed in the (M, cos ϑ) bin r. In addition to the 15 mass bins, the forward-backward asymmetry has two angular regions, cos ϑ ≥ 0, and cos ϑ < 0. Square transfer matrices for 30-element state vectors are implemented. The first 15 elements of the vectors are the mass bins for the cos ϑ ≥ 0 angular region, and the remainder for the other angular region. The simulation predicts significant bin-to-bin event migration among the mass bins when the produced and reconstructed values of cos ϑ have the same sign. For a mass bin, there is very little migration of events from one angular region to the other. As the simulation sample size is normalized to the integrated luminosity of the data, the transfer matrices provide properly normalized estimates of event migration between bins. An estimator for the true unfolding matrix is U¯gr = n¯gr/N¯r, where N¯r = g n¯gr is the expected total number of weighted events reconstructed in bin r. The 30-element state vector for N¯r is denoted as Nr, and the matrix U¯gr by U . The estimate for the resolution-unfolded state vector of produced events is Ng = U · Nr. For the event-weighting method, there are two transfer matrices that correspond to the weighted-event counts Nn and Nd of Eq. (5), and thus two separate unfolding matrices U , two separate event-weighted measurements of Nr, and two separate estimates of the resolution- unfolded Ng. The CC- and CP-event estimates of Ng for the numerator and denominator of Afb are summed prior to the evaluation of Afb. The measurements of Afb for the 15 mass bins are collectively denoted by Afb. The covariance matrix of the Afb measurement, de- noted by V , is calculated using the unfolding matrices, the expectation values of Nr and Afb from the simulation, and their fluctuations over an ensemble. The per- experiment fluctuation to Ng is U · (Nr + δNr), where δNr represents a fluctuation from the expectation Nr. The variation δAfb resulting from the Ng fluctuation is ensemble-averaged to obtain the covariance matrix Vlm = (δAfb)l(δAfb)m , where (δAfb)k (k = l and m) denotes the kth element of δAfb. Each element i of Nr undergoes independent, normally distributed fluctuations with a variance equal to the value expected for N¯i. Because N¯i is a sum of event weights, fluctuations of N¯i are quantified with the variance of its event weights. The two Nr vectors, the numerator vector and the denominator vector, have correlations. Elements i of the numerator and denominator vectors contain the same events, the only difference being that they have different event weights. To include this correlation, the event-count variations of elements i of the numerator and denominator δNr vectors are based on the same fluctuation from a normal distribution with unit rms dispersion. The covariance matrix is expanded and inverted to the error matrix using singular-value decomposition methods. As the covariance matrix is a real-valued symmetric 15 × 15 matrix, its 15 eigenvalues and eigenvectors are the rank-1 matrix components in the decomposition of the covariance matrix and the error matrix V = λn |vn vn| and n V −1 = λ−n 1 |vn vn| , n (6) where λn and |vn are the eigenvalues and eigenvectors of V , respectively, and |vn vn| represents a vector projection operator in the notation of Dirac bra-kets. The covariance matrix can have eigenvalues that are very small relative to the largest eigenvalue. Their vector projection operators select the fine structure of the resolution model, and at a small enough eigenvalue, they become particular to the simulation and include 20 True Afb -- estimated Afb Afb 0.02 First/last bin: underflow/overflow 0.01 0.6 First/last Bin: underflow/overflow Data PYTHIA POWHEG-BOX NLO 0.4 0 0.2 -0.01 -0.02 60 70 80 90 100 110 120 130 140 150 M (GeV/c2) 0 -0.2 FIG. 19. Event-weighting bias in bins of the electron-pair invariant mass. The biases are the crosses, and the uncertainties are the bin-by-bin unfolding estimates of the simulation. The superimposed histogram is the difference between the Afb calculations for the rapidity range |y| < 1.7 and |y| < 1.5, and the uncertainties are estimates for the PDF uncertainty. noise. While their contribution to the covariance matrix is small, they completely dominate the error matrix. The fine structures of the simulation, measurement, and calculation are different. Consequently, comparisons between the Afb measurement and predictions that use the error matrix are unstable. To alleviate these instabilities, the decomposition of the error matrix, Eq. (6), is regulated so that the contributions of eigenvectors with very small eigenvalues are suppressed. A general method, as described below, is to add a regularization term or function rn to the eigenvalues: λn → λn + rn, where λn + rn is the regularized eigenvalue. D. Event-weighting bias correction After resolution unfolding, the event-weighted Afb values have second-order acceptance and efficiency biases from regions of limited boson acceptance, and to a lesser extent, from detector nonuniformities resulting in ( A)+ = ( A)−. The bias is defined as the difference between the true value of Afb before QED FSR calculated with pythia and the unfolded simulation estimate. The size of the simulation sample is 21 times that of the data. The bias is a mass-bin-specific additive correction to the unfolded Afb measurement, and is shown in Fig. 19. All significant bias corrections are less than 8% of the magnitude of Afb and most of them are 3% or less. Figure 19 also shows the difference between asymmetries calculated with the measurement rapidity range |y| < 1.7, and with a reduced range |y| < 1.5. The difference is representative of contributions to the bias from regions of reduced acceptance at large values of |y|, and the PDF uncertainty of the difference is specified later in Sec. X A. For increasing values of |y|, there is a relative -0.4 60 70 80 90 100 110 120 130 140 150 M (GeV/c2) FIG. 20. Fully corrected Afb for electron pairs with |y| < 1.7. The measurement uncertainties are bin-by-bin unfolding estimates. The vertical line is M = MZ . The pythia calculation uses sin2 θeleffpt = 0.232. The EBA-based powheg-box calculation uses sin2 θW = 0.2243 (sin2 θeleffpt = 0.2325) and the default PDF of NNPDF-3.0. TABLE II. Fully corrected Afb measurement for electron pairs with |y| < 1.7. The measurement uncertainties are bin- by-bin unfolding estimates. Mass bin (GeV/c2) 50–64 64–74 74–80 80–84 84–86 86–88 88–90 90–92 92–94 94–96 96–100 100–108 108–120 120–150 150–350 Afb −0.262 ± 0.014 −0.409 ± 0.015 −0.348 ± 0.015 −0.224 ± 0.014 −0.134 ± 0.014 −0.068 ± 0.010 −0.0015 ± 0.0044 0.0533 ± 0.0017 0.1021 ± 0.0036 0.1570 ± 0.0087 0.2228 ± 0.0094 0.335 ± 0.010 0.473 ± 0.012 0.541 ± 0.012 0.597 ± 0.014 increase of the u- to d-quark flux and a decrease of the antiquark flux from the proton. The covariance matrix of the bias-correction uncertainties is combined with the covariance matrix for the Afb measurement. The fully corrected measurement of Afb, including the bias correction, is shown in Fig. 20 and tabulated in Table II. 21 IX. EXTRACTION OF sin2 θeleffpt The Drell-Yan asymmetry measurement is directly sensitive to the effective-mixing terms sin2 θeff , which are products of the form-factor functions with the static sin2 θW parameter (Sec. III A). The asymmetry is most sensitive to the value of the effective-leptonic sin2 θeff term in the vicinity of the Z pole, or sin2 θeleffpt, and its value is derived from the sin2 θW parameter of the Afb template that best describes the measurement. For nonEBA calculations such as pythia, the template parameter is sin2 θeleffpt. While the value of sin2 θeleffpt is a direct measurement, the interpretation of the corresponding value of sin2 θW and the form factors are dependent on the details of the EBA model. The measurement and templates are compared using the χ2 statistic evaluated with the Afb measurement error matrix. Each template corresponds to a particular value of sin2 θW and provides a scan point for the χ2 function: χ2(sin2 θW ). The χ2 values of the scan points are fit to a parabolic χ2 function, χ2(sin2 θW ) = χ¯2 + (sin2 θW − sin2θW )2/σ¯2 , (7) where χ¯2, sin2θW , and σ¯ are parameters. The sin2θW parameter is the best-fit value of sin2 θW , σ¯ is the corresponding measurement uncertainty, and χ¯2 is the associated goodness-of-fit between the Afb measurement and calculation over the 15 mass bins. Without regularization of the error matrix, there are large fluctuations of the χ2 values for each scan point from the expected parabolic form. Such fluctuations are induced by the small eigenvalue terms in the expansion of the error matrix, described in Eq. (6). To attenuate these fluctuations, the regularization function method described at the end of Sec. VIII C is used. The eigenvalues and regularization terms are shown in Fig. 21. The horizontal line of Fig. 21 is an estimate, detailed next, of the resolving power of the measurement. The eigenvectors whose eigenvalues are below the line tend to project simulation structure finer than the resolution of the measurement, and thus induce instabilities. The effectiveness of the regularization is measured with the goodness-of-fit between the χ2(sin2 θW ) value of the scan points and the parabolic function. In the basis vector space of the error matrix, the χ2 of each scan point is n(δAfb)2n/(λn + rn), where the index n runs over all the eigenvector numbers and δAfb is the difference between the measured and calculated values of Afb. The regularization function shown in Fig. 21 is defined and optimized as follows. The shape of the regularization function is chosen so that it selectively suppresses eigenvectors that project onto noise rather than the uncertainties of the measurement. To identify these eigenvectors, the expansion of the error matrix is truncated one eigenvector at a time. Truncating eigenvectors 14 and 13 from the error matrix significantly improves the goodness-offit. There is no further improvement with the truncation λn 10-4 10-5 10-6 10-7 10-8 10-9 0 2 4 6 8 10 12 14 n: Eigenvector number FIG. 21. Eigenvalues of the error matrix (solid histogram), and its regularization terms (dashed histogram). The horizontal line is the square of the statistical uncertainty of the Afb measurement for the mass bin containing the Z peak. of lower numbered eigenvectors. Consequently, the regularization terms for eigenvectors 13 and 14 are set to values significantly larger than the eigenvalues so that the contributions of their constituents to the χ2 are negligible. The regularization term for eigenvector 12 is set to a value that is comparable with its companion eigenvalue. For eigenvector numbers 11 and under, the regularization terms are set to zero or values much smaller than the eigenvalues so that their components in the χ2 are unaffected or negligibly affected by the regularization terms. The optimum normalization level is determined via a scan of level scale-factor values, starting from 0. As the scale-factor value increases, the goodness-of-fit rapidly improves then enters a plateau region without significant improvement and only a degradation of the measurement resolution. The optimum is chosen to be slightly beyond the start of the plateau region, where the sin2θW parameter is also stable in value. As a cross check, the extraction of sin2 θW is performed using only CC or CP events for the measurement of Afb and its error matrix, and the default PDF of NNPDF3.0 for the calculation of templates. The extracted values using only CC or CP events differ by about 0.6 standard deviations of the statistical uncertainty. Since they are consistent with each other, CC and CP events are hereafter combined. An example template scan extraction of sin2 θW from the asymmetry of CC and CP events using χ2 values calculated with the regularized error matrix, and then fit to the parabolic χ2 function of Eq. (7) is shown in Fig. 22. The EBA-based tree and powheg-box NLO calculations of Afb use NNPDF-3.0 PDFs, an ensemble of probability-based PDFs. Such ensembles are random samples drawn from the probability density distribution of PDF parameters constrained by a global fit to prior measurements. Thus, all information within the probability density distribution is utilized. The predicted 22 40 35 χ2 Afb−Afb(PYTHIA) χ 2 35 30 25 20 15 0.222 0.223 0.224 0.225 0.226 sin2θW FIG. 22. Values of χ2 as functions of scan points in the sin2 θW variable with the parabolic fit overlaid. The triangles are the comparisons of the electron-pair Afb measurement with the powheg-box NLO calculations. The Afb templates of each scan point are calculated with the default PDF of NNPDF-3.0. The solid curve is the fit of those points to the χ2 parabolic function. value of an observable is the convolution of the probability density distribution with the calculation. Consequently, the rms dispersion about the mean is the associated PDF uncertainty [46]. Typically, the PDF ensemble consists of equally likely samples. The NNPDF3.0 ensemble consists of 100 equally probable samples. New measurements, if compatible with the measurements used to constrain the PDFs, are incorporated into the ensemble without regenerating it. This is accomplished by weighting the ensemble PDFs, numbered 1 to N , with the likelihood of the new measurement being consistent with the calculations: wk = exp(− 1 2 χ2k ) N l=1 exp(− 1 2 χ2l ) (8) where wk is the weight for PDF number k, and χ2k is the χ2 between the new measurement and the calculation using that PDF [46, 47]. These weights are denoted as wk weights [46]. The Afb measurement is used simultaneously to ex- tract sin2 θeleffpt and to constrain PDFs [48]. Scan templates of Afb are calculated for each ensemble PDF, and its best-fit parameters, sin2θW , χ¯2 and σ¯, are derived. Figure 23 shows that the Afb measurement is compatible with those included in the NNPDF-3.0 fits of PDF parameters. The results of the template scans are sum- marized in Table III. Included in the table for comparison are other measurements of sin2 θeleffpt; the CDF results are derived from EBA-based QCD templates. The EBA-based powheg-box calculations of Afb using the wk-weighted PDFs give the central value of sin2 θW . The χ¯2 values listed in Table III indicate that the powheg-box calculation provides the best descrip- tion of the Afb measurement. For graphical compar- 30 25 20 15 0.2236 0.2239 0.2242 sin 2θW 0.2245 0.2248 FIG. 23. χ¯2 versus sin2θW parameters for the best-fit templates of the powheg-box NLO calculation for each of the NNPDF-3.0 ensemble PDFs. The Afb measurement with electron pairs covers 15 mass bins. 0.03 First/last bin: underflow/overflow 0.02 0.01 0 -0.01 -0.0260 70 80 90 100 110 120 130 140 150 M (GeV/c2) FIG. 24. Afb −Afb(pythia) for |y| < 1.7. The diamonds represent the measurement using electron pairs, and the uncer- tainties shown are the bin-by-bin unfolding estimates which are correlated. There are no suppressed measurement values. The solid bars represent the powheg-box calculation with the default NNPDF-3.0 PDFs. The dashed bars represent the resbos calculation with CTEQ6.6 PDFs. Both calculations use sin2 θW = 0.2243. The horizontal line represents the reference pythia calculation which uses CTEQ5L PDFs with sin2 θeleffpt = 0.232. isons of best-fit Afb templates, the difference relative to a reference calculation is used: Afb − Afb(pythia) where the reference Afb(pythia) is the tuned pythia calculation described in Sec. VI on the signal simulation. Figure 24 shows the difference distributions for the mea- surement, the powheg-box calculation, and the resbos calculation. The comparison of powheg-box with NNPDF-3.0 PDFs to resbos with CTEQ6.6 PDFs illustrates the nature of Afb(M ) as a simultaneous probe of the electroweak-mixing parameter and the PDFs. The NNPDF-3.0 PDFs include collider data from the LHC 23 TABLE III. Extracted values of sin2 θeleffpt and sin2 θW from the Afb measurement using electron pairs. For the tree and powheg-box entries, the values are averages over the NNPDF-3.0 ensemble; “weighted” templates denote the wk-weighted average; and δ sin2 θW is the PDF uncertainty. The pythia entry is the value from the scan over non-EBA templates calculated by pythia 6.4 with CTEQ5L PDFs. The uncertainties of the electroweak-mixing parameters are the measurement uncertainties σ¯. For the χ¯2 column, the number in parenthesis is the number of mass bins of the Afb measurement. Template (Measurement) powheg-box NLO, default powheg-box NLO, weighted resbos NLO Tree LO, default Tree LO, weighted pythia (CDF 9 fb−1 A(fbµµ) [6]) (CDF 2 fb−1 A(4ee) [5]) (LEP-1 and SLD A0F,Bb [11]) (SLD A [11]) sin2 θeleffpt 0.23249 ± 0.00049 0.23248 ± 0.00049 0.23249 ± 0.00049 0.23252 ± 0.00049 0.23250 ± 0.00049 0.23207 ± 0.00046 0.2315 ± 0.0010 0.2328 ± 0.0010 0.23221 ± 0.00029 0.23098 ± 0.00026 sin2 θW 0.22429 ± 0.00048 0.22428 ± 0.00048 0.22429 ± 0.00047 0.22432 ± 0.00047 0.22430 ± 0.00047 − 0.2233 ± 0.0009 0.2246 ± 0.0009 − − δ sin2 θW ±0.00020 ±0.00018 − ±0.00021 ±0.00021 − − − − − χ¯2 15.9 (15) 15.4 (15) 21.3 (15) 22.4 (15) 21.5 (15) 24.6 (15) 21.1 (16) − − − while the CTEQ6.6 PDFs do not. X. SYSTEMATIC UNCERTAINTIES The systematic uncertainties of the results derived from electron pairs contain contributions from both the measurement of Afb and the template predictions of Afb for various input values of sin2 θW . Both the experimental and prediction-related systematic uncertainties are small compared to the experimental statistical uncertainty. The Afb templates of the powheg-box calculations are used to estimate systematic uncertainties on the sin2 θW parameter from various sources. A. Measurement The measurement uncertainties considered are from the energy scale and resolution, and from the background estimates. The bias-correction uncertainty from the PDFs, expected to be a small secondary effect, is not included. For the propagation of uncertainties to the extracted value of sin2 θW , the default PDF of the NNPDF-3.0 ensemble is used. The total measurement systematic uncertainty is ±0.00003. The energy scale and resolution of the simulation and data samples are accurately calibrated (Sec. VII) using electron-pair mass distributions. In conjunction, the mass distributions of the simulation have been tuned to agree with those of the data, and the agreement between them, presented in Figs. 12 and 13 is good. Since the energy scales of the data and simulation are calibrated separately from the underlying-physics scale, the potential effect of an offset between the global scales of the simulation and data is investigated as a systematic uncertainty. The electron-pair mass distributions in the vicinity of the Z-boson mass peak are used to constrain shifts. Scale shifts for the central and plug EM calorimeters are considered separately. The resulting uncertainty from the energy scale is ±0.00003. The potential effect of the limitations to the energy-resolution model of the simulation is also investigated, and the resulting uncertainty is estimated to be negligible. For the background systematic uncertainty, the normalization uncertainties of the two largest backgrounds, QCD and Z → τ τ , are considered. They amount to about three-quarters of the total background. The uncertainties of their normalization values from the background fits described in Sec. VII C are propagated into uncertainties on sin2 θW . They have a negligible impact on the measurement. For the Z → τ τ background of the CC topology, the difference between the constrained and unconstrained fit normalizations is assigned as a systematic uncertainty. The systematic uncertainty from the background is ±0.00002. The bias correction uses the pythia calculation with CTEQ5L PDFs. To evaluate whether a PDF systematic uncertainty is needed, the following bias metric is used: the difference in asymmetries calculated with the measurement rapidity range of |y| < 1.7 and with the reduced rapidity range of |y| < 1.5. The bias metric calculated with pythia is shown in Fig. 19, along with the PDF uncertainties estimated using the tree-level calculation of Afb with the NNPDF-3.0 ensemble of PDFs. The PDF uncertainties are small when compared to the statistical uncertainties of the bias correction. In addition, the pythia calculation of the bias-metric function is compatible, relative to PDF uncertainties, with the tree-level calculation using NNPDF-3.0 PDFs; the comparison χ2 has a value of 11 for the 15 mass bins. The PDF uncertainty to the pythia calculation is not included with the measurement because its effects are sufficiently small relative to the statistical uncertainties of the bias correction, and because the prediction includes an uncertainty for PDFs. 24 sin2θW 0.2245 0.2244 0.2243 0.2242 0.2241 0.224 0.2239 0.2238 0.2237 0.232 0.2322 0.2324 sin2 θleefpf t 0.2326 0.2328 FIG. 25. sin2 θW versus sin2 θeleffpt relationships from zfitter calculations. The default calculation is the middle line of the group. The outermost lines are for one standard-deviation shifts to the default value of the top-quark mass parameter (173.2 ± 0.9) [49]; the lower line corresponds to a higher value of the top-quark mass. The lines for one standard-deviation variations of the ∆αe(5m)(MZ2 ) parameter are close to the default calculation and not easily distinguishable. The vertical line, an example reference value for sin2 θeleffpt, is explained in the text. B. Predictions The theoretical uncertainties considered are from the PDFs, higher-order QCD effects, and the zfitter calculation. The dominant uncertainty is the PDF uncertainty of ±0.00018, and it is the wk-weighted value of δ sin2 θW from the powheg-box NLO entry of Table III. The total prediction uncertainty is ±0.00020. The uncertainty of higher-order QCD effects is estimated with the difference between the values of sin2 θW in Table III extracted with the tree and powheg-box NLO templates based on the wk-weighted ensemble of NNPDF-3.0 PDFs. This uncertainty, denoted as the “QCD scale” uncertainty, is ±0.00002. Although the powheg-box prediction is a fixed-order NLO QCD calculation at large boson PT, it is a resummation calculation in the low-to-moderate PT region. The partonshowering algorithm of pythia incorporates multi-order real emissions of QCD radiation over all regions of the boson PT. The sin2 θeleffpt result, because of its direct relationship with Afb, is independent of the standard-model based calculations specified in the appendix. However, the choice of input parameter values may affect the fit value of sin2 θW or MW . The effect of measurement uncertainties from the top-quark mass mt and from the contribution of the light quarks to the “running” electromagnetic fine-structure constant at the Z mass ∆αe(5m)(MZ2 ) is investigated using these uncertainties: ±0.9 GeV/c2 [49] and ±0.0001 [50], respectively. Figure 25 shows the relation between sin2 θW and sin2 θeleffpt for the default pa- TABLE IV. Summary of the systematic uncertainties on the extraction of the electroweak-mixing parameters sin2 θeleffpt and sin2 θW from the Afb measurement with electron pairs. Source Energy scale Backgrounds NNPDF-3.0 PDF QCD scale Form factor sin2 θeleffpt ±0.00003 ±0.00002 ±0.00019 ±0.00002 − sin2 θW ±0.00003 ±0.00002 ±0.00018 ±0.00002 ±0.00008 rameter values, and for one standard-deviation shifts to the default values of the mt and ∆αe(5m)(MZ2 ) parameters. Offsets from the default parameter curve to the one standard-deviation curves along a reference value for sin2 θeleffpt (e.g., the vertical line in Fig. 25) are used as systematic uncertainties to sin2 θW from the input parameters. The uncertainty to sin2 θW from ∆αe(5m)(MZ2 ) is negligible, and that from mt is ±0.00008. This uncertainty, denoted as the “form factor” uncertainty, is included in systematic uncertainties for sin2 θW and MW . XI. RESULTS The values for sin2 θeleffpt and sin2 θW (MW ) extracted from this measurement of Afb are sin2 θeleffpt = 0.23248 ± 0.00049 ± 0.00019 sin2 θW = 0.22428 ± 0.00048 ± 0.00020 MW (indirect) = 80.313 ± 0.025 ± 0.010 GeV/c2 , where the first contribution to the uncertainties is statistical and the second is systematic. All systematic uncertainties are combined in quadrature. A summary of the sources and values of systematic uncertainties is presented in Table IV. The results of this section supersede those derived from the A4 angulardistribution coefficient of ee-pairs from a sample corresponding to 2.1 fb−1 of collisions [5]. XII. CDF RESULT COMBINATION The measurement of Afb presented in this paper and the previous CDF measurement using Drell-Yan µ+µ− pairs [6] are used to extract the combined result for the electroweak-mixing parameter. Both measurements are fully corrected and use the full Tevatron Run II data set. Since they are defined for different regions of the leptonpair rapidity, |yee| < 1.7 and |yµµ| < 1.0, each measurement is compared separately to Afb templates calculated with the rapidity restriction of the measurements, and the joint χ2 is used to extract the combined values for electroweak-mixing parameters sin2 θeleffpt and sin2 θW . 25 60 55 50 χ2 45 40 35 0.2233 0.2236 0.2239 sin 2θW 0.2242 0.2245 FIG. 26. χ¯2 versus sin2θW parameters of the µµ- and eechannel combination. The prediction templates are calculated with powheg-box NLO and each of the NNPDF-3.0 ensemble PDFs. The µµ- and ee-channel Afb measurements contain 16 and 15 mass bins, respectively. A. Method The templates for both measurements are calculated using the EBA-based powheg-box NLO framework and the NNPDF-3.0 PDF ensemble of this analysis. The corresponding tree-level templates are also calculated. The Afb templates for both the µµ- and ee-channel measurements are calculated in the same powheg-box or tree-level computational runs. Thus, they share common events and scan-point values of the sin2 θW parameter. The method for the extraction of sin2 θeleffpt from each measurement is unaltered. For each of the ensemble PDFs, the parabolic fits to χ2(sin2 θW ) shown in Eq. (7) from each measurement are combined to obtain the values of sin2θW , χ¯2, and σ¯. Figure 26 shows the χ¯2 and sin2θW parameters associated with each ensemble PDF. The corresponding table of fit parameters is provided as supplemental material [51]. The ensemble-averaged values of the individual channels, along with their combination, are shown in Table V. The wk-weighted averaging method with powheg-box NLO calculations is selected for the central value of the combination result. B. Systematic uncertainties The categories of systematic uncertainties for both the µµ- and ee-channel extractions of the electroweak-mixing parameters are the same. Uncertainties associated with the measurements include those on the electroweakmixing parameter from the backgrounds and the energy scales. Those associated with the predictions include uncertainties from the PDFs and higher-order QCD effects (QCD scale). The numerical values for systematic uncertainties in this section are for the sin2 θW parameter. The measurement uncertainties of the µµ and ee chan- nels are uncorrelated, and thus the propagation of their uncertainties to sin2 θW is uncorrelated. The combined energy-scale and background uncertainties are ±0.00002 and ±0.00003, respectively. As the prediction uncertainties of both channels are correlated, the corresponding uncertainties of the combination are derived from the fit parameters of the joint χ2. The uncertainty due to the PDF is ±0.00016, which is the wk-weighted δ sin2 θW value from the powheg-box NLO entry of Table V. The uncertainty due to the QCD scale is ±0.00007, which is the difference between the wkweighted sin2 θW values of the powheg-box NLO and Tree entries from Table V. C. Results The combination values for sin2 θeleffpt and sin2 θW (MW ) are sin2 θeleffpt = 0.23221 ± 0.00043 ± 0.00018 sin2 θW = 0.22400 ± 0.00041 ± 0.00019 MW (indirect) = 80.328 ± 0.021 ± 0.010 GeV/c2 , where the first contribution to the uncertainties is statistical and the second is systematic. All systematic uncertainties are combined in quadrature, and the sources and values of these uncertainties are summarized in Table VI. The form-factor uncertainty, estimated in Sec. XI, is the uncertainty from the standard-model based calculation specified in the appendix. The measurements of sin2 θeleffpt are compared with previous results from the Tevatron, LHC, LEP-1, and SLC in Fig. 27. The hadron collider results are based on Afb measurements. The LEP-1 and SLD results on sin2 θeleffpt are from these asymmetry measurements at the Z pole [11]: A0F,B → 0.23099 ± 0.00053 A (Pτ ) → 0.23159 ± 0.00041 A (SLD) → 0.23098 ± 0.00026 A0F,Bb → 0.23221 ± 0.00029 AF0,Bc → 0.23220 ± 0.00081 QhFaBd → 0.2324 ± 0.0012 The QhFaBd measurement is based on the hadronic-charge asymmetry from all-hadronic final states. The W -boson mass inference is compared in Fig. 28 with previous direct and indirect measurements from the Tevatron, NuTeV, LEP-1, SLD, and LEP-2. The direct measurement is from the Tevatron and LEP-2 [52]. The previous indirect measurement from the Tevatron is derived from the CDF measurement of Afb with muon pairs and it uses the same EBA-based method of inference. The indirect measurement of sin2 θW from LEP-1 and SLD, 0.22332±0.00039, is from the standard-model fit to all Z-pole measurements [11, 12] described in Appendix F 26 TABLE V. Extracted values of sin2 θeleffpt and sin2 θW after averaging over the NNPDF-3.0 ensembles. The “weighted” templates denote the wk-weighted ensembles; and δ sin2 θW is the PDF uncertainty. The uncertainties of the electroweak-mixing parameters are the measurement uncertainties σ¯. For the χ¯2 column, the number in parenthesis is the number of mass bins of the Afb measurement. The ee-channel values are from Table III, and the µµ-channel values use the previous CDF measurement of Afb with µ+µ− pairs [6]. Template powheg-box NLO, default powheg-box NLO, weighted powheg-box NLO, default powheg-box NLO, weighted powheg-box NLO, default powheg-box NLO, weighted Tree LO, default Tree LO, weighted Tree LO, default Tree LO, weighted Tree LO, default Tree LO, weighted Channel µµ µµ ee ee ee + µµ ee + µµ µµ µµ ee ee ee + µµ ee + µµ sin2 θeleffpt 0.23140 ± 0.00086 0.23141 ± 0.00086 0.23249 ± 0.00049 0.23248 ± 0.00049 0.23222 ± 0.00043 0.23221 ± 0.00043 0.23154 ± 0.00085 0.23153 ± 0.00085 0.23252 ± 0.00049 0.23250 ± 0.00049 0.23228 ± 0.00042 0.23215 ± 0.00043 sin2 θW 0.22316 ± 0.00083 0.22317 ± 0.00083 0.22429 ± 0.00048 0.22428 ± 0.00048 0.22401 ± 0.00041 0.22400 ± 0.00041 0.22330 ± 0.00082 0.22329 ± 0.00082 0.22432 ± 0.00047 0.22430 ± 0.00047 0.22407 ± 0.00041 0.22393 ± 0.00041 δ sin2 θW ±0.00029 ±0.00028 ±0.00020 ±0.00018 ±0.00021 ±0.00016 ±0.00031 ±0.00029 ±0.00021 ±0.00021 ±0.00023 ±0.00016 χ¯2 21.0 (16) 20.7 (16) 15.9 (15) 15.4 (15) 38.3 (31) 35.9 (31) 20.9 (16) 20.5 (16) 22.4 (15) 21.5 (15) 44.4 (31) 37.4 (31) TABLE VI. Summary of the systematic uncertainties on the µµ- and ee-channel combination for the electroweak-mixing parameters sin2 θeleffpt and sin2 θW . Source Energy scale Backgrounds NNPDF-3.0 PDF QCD scale Form factor sin2 θeleffpt ±0.00002 ±0.00003 ±0.00016 ±0.00006 − sin2 θW ±0.00002 ±0.00003 ±0.00016 ±0.00007 ±0.00008 of Ref. [12]. The following input parameters to zfitter, the Higgs-boson mass mH , the Z-boson mass MZ , the QCD coupling at the Z pole αs(MZ2 ), and the QED correction ∆αe(5m)(MZ2 ), are varied simultaneously within the constraints of the LEP-1 and SLD data, while the topquark mass mt is constrained to the directly measured value from the Tevatron, 173.2 ± 0.9 GeV/c2 [49]. The NuTeV value is an inference based on the on-shell sin2 θW parameter extracted from the measurement of the ratios of the neutral-to-charged current ν and ν¯ cross sections at Fermilab [13]. XIII. SUMMARY The angular distribution of Drell-Yan lepton pairs provides information on the electroweak-mixing parameter sin2 θW . The electron forward-backward asymmetry in the polar-angle distribution cos ϑ is governed by the A4 cos ϑ term, whose A4 coefficient is directly related to the sin2 θeleffpt mixing parameter at the lepton vertex, and indirectly to sin2 θW . The effective-leptonic parameter sin2 θeleffpt is derived from the measurement of the forwardbackward asymmetry Afb(M ) based on the entire CDF Run II sample of electron pairs, reconstructed in 9.4 fb−1 of integrated luminosity from pp¯ collisions at a center-ofmomentum energy of 1.96 TeV. Calculations of Afb(M ) with different values of the electroweak-mixing parameter are compared with the measurement to determine the value of the parameter that best describes the data. The calculations include QCD radiative corrections and virtual electroweak radiative corrections. For the ee-channel measurement of Afb presented in this paper, the best-fit values from the comparisons are sin2 θeleffpt = 0.23248 ± 0.00053, sin2 θW = 0.22428 ± 0.00051, and MW (indirect) = 80.313 ± 0.027 GeV/c2 . Each uncertainty includes statistical and systematic contributions. The inferred value of sin2 θW (MW ) is based on the standard-model calculations specified in the appendix. When this measurement of Afb and the previous CDF measurement based on muon pairs [6] are used jointly in fits, the corresponding best-fit values are sin2 θeleffpt = 0.23221 ± 0.00046, sin2 θW = 0.22400 ± 0.00045, and MW (indirect) = 80.328 ± 0.024 GeV/c2 . Both results are consistent with LEP-1 and SLD measurements at the Z-boson pole. The value of sin2 θeleffpt is also consistent with the previous results from the Tevatron [6, 7]. ACKNOWLEDGMENTS We thank the Fermilab staff and the technical staffs of the participating institutions for their vital contributions. This work was supported by the U.S. Department of Energy and National Science Foundation; the Italian Istituto Nazionale di Fisica Nucleare; the Ministry of 27 10.5 LEP-1 and SLD: Z-pole LEP-1 and SLD: A0F,Bb SLD: Al CMS µµ 1 fb-1 0.23149±0.00016 0.23221±0.00029 0.23098±0.00026 ATLAS ee+µµ 5 fb-1 0.2287±0.0032 0.2308±0.0012 LHCb µµ 3 fb-1 0.23142±0.00107 D0 ee 10 fb-1 0.23147±0.00047 CDF µµ 9 fb-1 0.2315±0.0010 CDF ee 9 fb-1 0.23248±0.00053 CDF ee+µµ 9 fb-1 0.23221±0.00046 0 0.226 0.228 0.23 0.232 0.234 sin2 θleefpf t FIG. 27. Comparison of experimental measurements of sin2 θeleffpt. The horizontal bars represent total uncertainties. The CDF µµ-channel, ee-channel, and combination results are denoted as CDF µµ 9 fb−1 [6], CDF ee 9 fb−1, and CDF ee + µµ 9 fb−1, respectively. The other measurements are LEP-1 and SLD [11], CMS [9], ATLAS [8], LHCb [10], and D0 [7] The LEP-1 and SLD Z pole result is the combination of their six measurements. Education, Culture, Sports, Science and Technology of Japan; the Natural Sciences and Engineering Research Council of Canada; the National Science Council of the Republic of China; the Swiss National Science Foundation; the A.P. Sloan Foundation; the Bundesministerium fu¨r Bildung und Forschung, Germany; the Korean World Class University Program, the National Research Foundation of Korea; the Science and Technology Facilities Council and the Royal Society, United Kingdom; the Russian Foundation for Basic Research; the Ministerio de Ciencia e Innovaci´on, and Programa Consolider-Ingenio 2010, Spain; the Slovak R&D Agency; the Academy of Finland; the Australian Research Council (ARC); and the EU community Marie Curie Fellowship Contract No. 302103. Appendix: ZFITTER The input parameters to the zfitter radiativecorrection calculation are particle masses, the electromagnetic fine-structure constant αem, the Fermi constant GF , the strong-interaction coupling at the Z mass αs(MZ2 ), and the contribution of the light quarks to the 12 Indirect measurements LEP-1 and SLD (mt) NuTeV CDF µ µ 9 fb-1 CDF ee 9 fb-1 CDF ee+µ µ 9 fb-1 80.363±0.020 80.135±0.085 80.365±0.047 80.313±0.027 80.328±0.024 Direct measurement TeV and LEP-2 80.385±0.015 0 80 80.1 80.2 80.3 80.4 80.5 80.6 W-boson mass (GeV/c2) FIG. 28. Comparison of experimental determinations of the W -boson mass. The horizontal bars represent total uncer- tainties. The CDF µµ-channel, ee-channel, and combination results are denoted as CDF µµ 9 fb−1 [6], CDF ee 9 fb−1, and CDF ee+µµ 9 fb−1, respectively. The other indirect measure- ments are from LEP-1 and SLD [11, 12], which include the Tevatron top-quark mass measurement [49], and NuTeV [13]. The direct measurement is from the Tevatron and LEP-2 [52]. “running” αem at the Z mass ∆αe(5m)(MZ2 ). The scaledependent couplings are αs(MZ2 ) = 0.118 ± 0.001 [53] and ∆αe(5m)(MZ2 ) = 0.0275 ± 0.0001 [50]. The mass parameters are MZ = 91.1875 ± 0.0021 GeV/c2 [11, 12], mt = 173.2±0.9 GeV/c2 (top quark) [49], and mH = 125 GeV/c2 (Higgs boson). Form factors and the Z-boson total decay-width ΓZ are calculated. The central values of the parameters provide the context of the zfitter standard-model calculations. zfitter uses the on-shell renormalization scheme scheme [3], where particle masses are on-shell and sin2 θW = 1 − MW2 /MZ2 (A.1) holds to all orders of perturbation theory by definition. If both GF and mH are specified, sin θW is not independent, and related to GF and mH by standardmodel constraints from radiative corrections. To vary the sin θW (MW ) parameter, the value of GF is not constrained. The value of the MW is varied over 80.0–80.5 GeV/c2, and for each value, zfitter calculates GF and the form factors. Each set of calculations corresponds to a family of physics models with standard-model like couplings where sin2 θW and the GF coupling are defined by the MW parameter. The Higgs-boson mass constraint mH = 125 GeV/c2 keeps the form factors within the vicinity of standard-model fit values from LEP-1 and SLD [11, 12]. 28 The primary purpose of zfitter is to provide tables of form factors for each model. As the form factors are calculated in the massless-fermion approximation, they only depend on the fermion weak isospin and charge, and are distinguished via three indices: e (electron type), u (up-quark type), and d (down-quark type). For the ee → Z → qq¯ process, the zfitter scatteringamplitude ansatz is √ Aq = i 4 2GF MZ2 sˆ − (MZ2 − i sˆΓZ /MZ ) 4T3eT3q ρeq [ e¯|γµ(1 + γ5)|e q¯|γµ(1 + γ5)|q + −4|Qe|κe sin2 θW e¯|γµ|e q¯|γµ(1 + γ5)|q + −4|Qq|κq sin2 θW e¯|γµ(1 + γ5)|e q¯|γµ|q + 16|QeQq|κeq sin4 θW e¯|γµ|e q¯|γµ|q ] , where q = u or d, the ρeq, κe, κq, and κeq are complex- valued form factors, the bilinear γ matrix terms are co- variantly contracted, and 1 2 (1 + γ5) is the left-handed he- licity projector in the zfitter convention. The κe form factors of the Au and Ad amplitudes are not equivalent; however, at sˆ = MZ2 , they are numerically equal. The ρeq, κe, and κq form factors are incorporated into QCD calculations as corrections to the Born-level gAf and gVf couplings, gVf → √ ρeq (T3f − 2Qf κf sin2 θW ) and gAf → √ ρeq T3f , where f = e or q. The resulting current-current amplitude is similar to Aq, but the sin4 θW term contains κeκq. 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