PHYSICAL REVIEW Ddigital.csic.es/bitstream/10261/110540/1/Indirect measurement of si… · Indirect...

Indirect measurement of sin2θW (orMW) using μþμ− pairs from γ�=Z bosonsproduced in pp collisions at a center-of-momentum energy of 1.96 TeV

T. Aaltonen,21 S. Amerio,39b,39a D. Amidei,31 A. Anastassov,15,w A. Annovi,17 J. Antos,12 G. Apollinari,15 J. A. Appel,15

T. Arisawa,52 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. Barria,41a P. Bartos,12 M. Bauce,39b,39a F. Bedeschi,41a S. Behari,15

G. Bellettini,41b,41a J. Bellinger,54 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. Brigliadori,6b,6a C. Bromberg,32 E. Brucken,21 J. Budagov,13

H. S. Budd,44 K. Burkett,15 G. Busetto,39b,39a P. Bussey,19 P. Butti,41b,41a A. Buzatu,19 A. Calamba,10 S. Camarda,4

M. Campanelli,28 F. Canelli,11,dd B. Carls,22 D. Carlsmith,54 R. Carosi,41a S. Carrillo,16,m B. Casal,9,k M. Casarsa,48a

A. Castro,6b,6a P. Catastini,20 D. Cauz,48b,48c,48a V. Cavaliere,22 M. Cavalli-Sforza,4 A. Cerri,26,f L. Cerrito,28,r Y. C. Chen,1

M. Chertok,7 G. Chiarelli,41a G. Chlachidze,15 K. Cho,25 D. Chokheli,13 A. Clark,18 C. Clarke,53 M. E. Convery,15

J. Conway,7 M. Corbo,15,z M. Cordelli,17 C. A. Cox,7 D. J. Cox,7 M. Cremonesi,41a D. Cruz,47 J. Cuevas,9,y R. Culbertson,15

N. d’Ascenzo,15,v M. Datta,15,gg P. de Barbaro,44 L. Demortier,45 M. Deninno,6a M. D’Errico,39b,39a F. Devoto,21

A. Di Canto,41b,41a B. Di Ruzza,15,q J. R. Dittmann,5 S. Donati,41b,41a M. D’Onofrio,27 M. Dorigo,48d,48a A. Driutti,48b,48c,48a

K. Ebina,52 R. Edgar,31 A. Elagin,47 R. Erbacher,7 S. Errede,22 B. Esham,22 S. Farrington,38 J. P. Fernández Ramos,29

R. Field,16 G. Flanagan,15,t R. Forrest,7 M. Franklin,20 J. C. Freeman,15 H. Frisch,11 Y. Funakoshi,52 C. Galloni,41b,41a

A. F. Garfinkel,43 P. Garosi,41c,41a H. Gerberich,22 E. Gerchtein,15 S. Giagu,46a V. Giakoumopoulou,3 K. Gibson,42

C. M. Ginsburg,15 N. Giokaris,3 P. Giromini,17 G. Giurgiu,23 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. González López,29 I. Gorelov,34 A. T. Goshaw,14

K. Goulianos,45 E. Gramellini,6a S. Grinstein,4 C. Grosso-Pilcher,11 R. C. Group,51,15 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,53 T. Harrington-Taber,15,n K. Hatakeyama,5 C. Hays,38

J. Heinrich,40 M. Herndon,54 A. Hocker,15 Z. Hong,47 W. Hopkins,15,g S. Hou,1 R. E. Hughes,35 U. Husemann,55

M. Hussein,32,bb J. Huston,32 G. Introzzi,41e,41f,41a M. Iori,46b,46a A. Ivanov,7,p 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,53 A. Kasmi,5 Y. Kato,37,o W. Ketchum,11,hh J. Keung,40 B. Kilminster,15,dd D. H. Kim,25 H. S. Kim,25

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,52 M. Kirby,15 K. Knoepfel,15

K. Kondo,52,a 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. Lannon,35,x G. Latino,41c,41a H. S. Lee,25 J. S. Lee,25 S. Leo,41a

S. Leone,41a J. D. Lewis,15 A. Limosani,14,s E. Lipeles,40 A. Lister,18,b H. Liu,51 Q. Liu,43 T. Liu,15 S. Lockwitz,55

A. Loginov,55 D. Lucchesi,39b,39a A. Lucà,17 J. Lueck,24 P. Lujan,26 P. Lukens,15 G. Lungu,45 J. Lys,26 R. Lysak,12,e

R. Madrak,15 P. Maestro,41c,41a S. Malik,45 G. Manca,27,c A. Manousakis-Katsikakis,3 L. Marchese,6a,ii F. Margaroli,46a

P. Marino,41d,41a M. Martínez,4 K. Matera,22 M. E. Mattson,53 A. Mazzacane,15 P. Mazzanti,6a R. McNulty,27,j 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,6a C. S. Moon,15,z

R. Moore,15,ee,ff M. J. Morello,41d,41a A. Mukherjee,15 Th. Muller,24 P. Murat,15 M. Mussini,6b,6a J. Nachtman,15,n Y. Nagai,49

J. Naganoma,52 I. Nakano,36 A. Napier,50 J. Nett,47 C. Neu,51 T. Nigmanov,42 L. Nodulman,2 S. Y. Noh,25 O. Norniella,22

L. Oakes,38 S. H. Oh,14 Y. D. Oh,25 I. Oksuzian,51 T. Okusawa,37 R. Orava,21 L. Ortolan,4 C. Pagliarone,48a E. Palencia,9,f

P. Palni,34 V. Papadimitriou,15 W. Parker,54 G. Pauletta,48b,48c,48a M. Paulini,10 C. Paus,30 T. J. Phillips,14 G. Piacentino,41a

E. Pianori,40 J. Pilot,7 K. Pitts,22 C. Plager,8 L. Pondrom,54 S. Poprocki,15,g K. Potamianos,26 A. Pranko,26 F. Prokoshin,13,aa

F. Ptohos,17,h G. Punzi,41b,41a N. Ranjan,43 I. Redondo Fernández,29 P. Renton,38 M. Rescigno,46a F. Rimondi,6a,a

L. Ristori,41a,15 A. Robson,19 T. Rodriguez,40 S. Rolli,50,i M. Ronzani,41b,41a R. Roser,15 J. L. Rosner,11 F. Ruffini,41c,41a

A. Ruiz,9 J. Russ,10 V. Rusu,15 W. K. Sakumoto,44 Y. Sakurai,52 L. Santi,48b,48c,48a K. Sato,49 V. Saveliev,15,v

A. Savoy-Navarro,15,z P. Schlabach,15 E. E. Schmidt,15 T. Schwarz,31 L. Scodellaro,9 F. Scuri,41a S. Seidel,34 Y. Seiya,37

A. Semenov,13 F. Sforza,41b,41a S. Z. Shalhout,7 T. Shears,27 P. F. Shepard,42 M. Shimojima,49,u 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. Stentz,15,w 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. Thom,15,g E. Thomson,40 V. Thukral,47 D. Toback,47 S. Tokar,12 K. Tollefson,32 T. Tomura,49

D. Tonelli,15,f S. Torre,17 D. Torretta,15 P. Totaro,39a M. Trovato,41d,41a F. Ukegawa,49 S. Uozumi,25 F. Vázquez,16,m

G. Velev,15 C. Vellidis,15 C. Vernieri,41d,41a M. Vidal,43 R. Vilar,9 J. Vizán,9,cc M. Vogel,34 G. Volpi,17 P. Wagner,40

R. Wallny,15,k S. M. Wang,1 D. Waters,28 W. C. Wester III,15 D. Whiteson,40,d A. B. Wicklund,2 S. Wilbur,7

H. H. Williams,40 J. S. Wilson,31 P. Wilson,15 B. L. Winer,35 P. Wittich,15,g 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. Yi,15,n J. Yoh,15

K. Yorita,52 T. Yoshida,37,l G. B. Yu,14 I. Yu,25 A. M. Zanetti,48a Y. Zeng,14 C. Zhou,14 and S. Zucchelli6b,6a

(CDF Collaboration)

PHYSICAL REVIEW D 89, 072005 (2014)

1550-7998=2014=89(7)=072005(21) 072005-1 © 2014 American Physical Society

1Institute of Physics, Academia Sinica, Taipei, Taiwan 11529, Republic of China2Argonne National Laboratory, Argonne, Illinois 60439, USA

3University of Athens, 157 71 Athens, Greece4Institut de Fisica d’Altes Energies, ICREA, Universitat Autonoma de Barcelona,

E-08193 Bellaterra (Barcelona), Spain5Baylor University, Waco, Texas 76798, USA

6aIstituto Nazionale di Fisica Nucleare Bologna, I-40127 Bologna, Italy6bUniversity of Bologna, I-40127 Bologna, Italy

7University of California, Davis, Davis, California 95616, USA8University of California, Los Angeles, Los Angeles, California 90024, USA

9Instituto de Fisica de Cantabria, CSIC-University of Cantabria, 39005 Santander, Spain10Carnegie Mellon University, Pittsburgh, Pennsylvania 15213, USA

11Enrico Fermi Institute, University of Chicago, Chicago, Illinois 60637, USA12Institute of Experimental Physics, Comenius University,842 48 Bratislava, Slovakia; 040 01 Kosice, Slovakia

13Joint Institute for Nuclear Research, RU-141980 Dubna, Russia14Duke University, Durham, North Carolina 27708, USA

15Fermi National Accelerator Laboratory, Batavia, Illinois 60510, USA16University of Florida, Gainesville, Florida 32611, USA

17Laboratori Nazionali di Frascati, Istituto Nazionale di Fisica Nucleare, I-00044 Frascati, Italy18University of Geneva, CH-1211 Geneva 4, Switzerland

19Glasgow University, Glasgow G12 8QQ, United Kingdom20Harvard University, Cambridge, Massachusetts 02138, USA

21University of Helsinki,Helsinki FIN-00014, Finland and Helsinki Institute of Physics,

Helsinki FIN-00014, Finland22University of Illinois, Urbana, Illinois 61801, USA

23The Johns Hopkins University, Baltimore, Maryland 21218, USA24Institut für Experimentelle Kernphysik, Karlsruhe Institute of Technology,

D-76131 Karlsruhe, Germany25Center 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,

and Ewha Womans University, Seoul 120-750, Korea26Ernest Orlando Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA

27University of Liverpool, Liverpool L69 7ZE, United Kingdom28University College London, London WC1E 6BT, United Kingdom

29Centro de Investigaciones Energeticas Medioambientales y Tecnologicas, E-28040 Madrid, Spain30Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA

31University of Michigan, Ann Arbor, Michigan 48109, USA32Michigan State University, East Lansing, Michigan 48824, USA

33Institution for Theoretical and Experimental Physics (ITEP), Moscow 117259, Russia34University of New Mexico, Albuquerque, New Mexico 87131, USA

35The Ohio State University, Columbus, Ohio 43210, USA36Okayama University, Okayama 700-8530, Japan37Osaka City University, Osaka 558-8585, Japan

38University of Oxford, Oxford OX1 3RH, United Kingdom39aIstituto Nazionale di Fisica Nucleare, Sezione di Padova, I-35131 Padova, Italy

39bUniversity of Padova, I-35131 Padova, Italy40University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA

41aIstituto Nazionale di Fisica Nucleare Pisa, I-56127 Pisa, Italy41bUniversity of Pisa, I-56127 Pisa, Italy41cUniversity of Siena, I-56127 Pisa, Italy

41dScuola Normale Superiore, I-56127 Pisa, Italy41eINFN Pavia, I-27100 Pavia, Italy

41fUniversity of Pavia, I-27100 Pavia, Italy42University of Pittsburgh, Pittsburgh, Pennsylvania 15260, USA

43Purdue University, West Lafayette, Indiana 47907, USA

T. AALTONEN et al. PHYSICAL REVIEW D 89, 072005 (2014)

072005-2

44University of Rochester, Rochester, New York 14627, USA45The Rockefeller University, New York, New York 10065, USA

46aIstituto Nazionale di Fisica Nucleare, Sezione di Roma 1, I-00185 Roma, Italy46bSapienza Università di Roma, I-00185 Roma, Italy

47Mitchell Institute for Fundamental Physics and Astronomy, Texas A&M University,College Station, Texas 77843, USA

48aIstituto Nazionale di Fisica Nucleare Trieste, I-33100 Udine, Italy48bGruppo Collegato di Udine, I-33100 Udine, Italy

48cUniversity of Udine, I-33100 Udine, Italy48dUniversity of Trieste, I-34127 Trieste, Italy

49University of Tsukuba, Tsukuba, Ibaraki 305, Japan50Tufts University, Medford, Massachusetts 02155, USA

51University of Virginia, Charlottesville, Virginia 22906, USA52Waseda University, Tokyo 169, Japan

53Wayne State University, Detroit, Michigan 48201, USA54University of Wisconsin, Madison, Wisconsin 53706, USA

55Yale University, New Haven, Connecticut 06520, USA(Received 10 February 2014; published 4 April 2014)

Drell-Yan lepton pairs are produced in the process pp → μþμ− þ X through an intermediate γ�=Zboson.The forward-backward asymmetry in the polar-angle distribution of the μ− as a function of the invariantmass of the μþμ− pair is used to obtain the effective leptonic determination sin2θlepteff of the electroweak-mixing parameter sin2 θW, from which the value of sin2 θW is derived assuming the standard model. Themeasurement sample, recorded by the Collider Detector at Fermilab (CDF), corresponds to 9.2 fb−1 ofintegrated luminosity from pp collisions at a center-of-momentum energy of 1.96 TeV, and is the full CDFRun II data set. The value of sin2 θlepteff is found to be 0.2315� 0.0010, where statistical and systematicuncertainties are combined in quadrature. When interpreted within the context of the standard model using

aDeceased.bVisitor from University of British Columbia, Vancouver, BC V6T 1Z1, Canada.cVisitor from Istituto Nazionale di Fisica Nucleare, Sezione di Cagliari, 09042 Monserrato, Cagliari, Italy.dVisitor from University of California Irvine, Irvine, CA 92697, USA.eVisitor from Institute of Physics, Academy of Sciences of the Czech Republic, 182 21, Czech Republic.fVisitor from CERN, CH-1211 Geneva, Switzerland.gVisitor from Cornell University, Ithaca, NY 14853, USA.hVisitor from University of Cyprus, Nicosia CY-1678, Cyprus.iVisitor from Office of Science, U.S. Department of Energy, Washington, D.C. 20585, USA.jVisitor from University College Dublin, Dublin 4, Ireland.kVisitor from ETH, 8092 Zürich, Switzerland.lVisitor from University of Fukui, Fukui City, Fukui Prefecture, 910-0017 Japan.mVisitor from Universidad Iberoamericana, Lomas de Santa Fe, Distrito Federal C.P. 01219, México.nVisitor from University of Iowa, Iowa City, IA 52242, USA.oVisitor from Kinki University, Higashi-Osaka City, 577-8502 Japan.pVisitor from Kansas State University, Manhattan, KS 66506, USA.qVisitor from Brookhaven National Laboratory, Upton, NY 11973, USA.rVisitor from Queen Mary, University of London, London E1 4NS, United Kingdom.sVisitor from University of Melbourne, Victoria 3010, Australia.tVisitor from Muons, Inc., Batavia, IL 60510, USA.uVisitor from Nagasaki Institute of Applied Science, Nagasaki 851-0193, Japan.vVisitor from National Research Nuclear University, Moscow 115409, Russia.wVisitor from Northwestern University, Evanston, IL 60208, USA.xVisitor from University of Notre Dame, Notre Dame, IN 46556, USA.yVisitor from Universidad de Oviedo, E-33007 Oviedo, Spain.zVisitor from CNRS-IN2P3, Paris, F-75205 France.aaVisitor from Universidad Tecnica Federico Santa Maria, 110v Valparaiso, Chile.bbVisitor from The University of Jordan, Amman 11942, Jordan.ccVisitor from Universite catholique de Louvain, 1348 Louvain-La-Neuve, Belgium.ddVisitor from University of Zürich, 8006 Zürich, Switzerland.eeVisitor from Massachusetts General Hospital, Boston, MA 02114 USA.ffVisitor from Harvard Medical School, Boston, MA 02114 USA.ggVisitor from Hampton University, Hampton, VA 23668, USA.hhVisitor from Los Alamos National Laboratory, Los Alamos, NM 87544, USA.iiVisitor from Università degli Studi di Napoli Federico I, I-80138 Napoli, Italy.

INDIRECT MEASUREMENT OF sin2 θW (OR … PHYSICAL REVIEW D 89, 072005 (2014)

072005-3

the on-shell renormalization scheme, where sin2θW ¼ 1 −M2W=M

2Z, the measurement yields

sin2 θW ¼ 0.2233� 0.0009, or equivalently a W-boson mass of 80.365� 0.047 GeV=c2. The value ofthe W-boson mass is in agreement with previous determinations in electron-positron collisions and at theTevatron collider.

DOI: 10.1103/PhysRevD.89.072005 PACS numbers: 12.15.Lk, 13.85.Qk, 14.70.Hp

I. INTRODUCTION

In this paper, the angular distribution of charged leptonsðl�Þ from the Drell-Yan [1] process is used to measure theelectroweak-mixing parameter sin2 θW [2]. At the FermilabTevatron, Drell-Yan pairs are produced by the processpp → lþl− þ X, where the lþl− pair is producedthrough an intermediate γ�=Z boson, and X is the hadronicfinal state associated with the production of the boson. Inthe standard model, the production of Drell-Yan leptonpairs at the Born level proceeds through two parton-levelprocesses,

qq → γ� → lþl− and qq → Z → lþl−:

where the q and q are the quark and antiquark, respectively,from the colliding hadrons. The virtual photon couples thevector currents of the incoming and outgoing fermions ðfÞ,and the spacetime structure of a photon-fermion interactionvertex is hfjQfγμjfi, where Qf, the strength of thecoupling, is the fermion charge (in units of e), and jfiis the spinor for fermion f. An interaction vertex of afermion with a Z boson contains both vector ðVÞ and axial-vector ðAÞ current components, and its structure ishfjgfVγμ þ gfAγμγ5jfi. The Born-level coupling strengths are

gfV ¼ Tf3 − 2Qfsin2θW and gfA ¼ Tf

3 ;

where Tf3 is the third component of the fermion weak

isospin, which is Tf3 ¼ 1

2ð− 1

2Þ for positively (negatively)

charged fermions. At the Born level, and in all orders of theon-shell renormalization scheme, the sin2 θW parameter isrelated to theW-boson massMW and the Z-boson massMZ

by the relationship sin2 θW ¼ 1 −M2W=M

2Z. Weak-

interaction radiative corrections alter the strength of theBorn-level couplings into effective couplings. These effec-tive couplings have been investigated at the Tevatron [3–5],at the LHC [6], and at Large Electron-Positron Collider-1(LEP-1) and Stanford Linear Accelerator Center LargeDetector (SLD) [7]. Similar couplings have been inves-tigated with neutrino-nucleon collisions at the Tevatron [8]and with electron-proton collisions at Hadron ElectronRing Accelerator [9].The effective sin2 θW coupling at the lepton vertex,

denoted as sin2 θlepteff , has been accurately measured at theLEP-1 and SLD eþe− colliders. The combined average

of six individual measurements yields a value of0.23153� 0.00016 [7]. However, there is tension betweenthe two most precise individual measurements: the com-bined LEP-1 and SLD b-quark forward-backward asym-metry (A0;b

FBÞ yields sin2 θlepteff ¼ 0.23221� 0.00029, andthe SLD polarized left-right asymmetry ðAlÞ yieldssin2 θlepteff ¼ 0.23098� 0.00026. They differ by 3.2 stan-dard deviations.The Drell-Yan process at hadron-hadron colliders is also

sensitive to the sin2 θlepteff coupling. Measurements of theforward-backward asymmetry in the l− polar angle distri-bution as a function of the lepton-pair invariant mass are usedto extract the coupling. This paper presents a new measure-ment of the sin2 θlepteff coupling and an inference of the sin2 θWparameter using a sample of μþμ− pairs corresponding to anintegrated luminosity of 9.2 fb−1 collected at the Tevatron ppcollider. Innovative methods for the calibration of the muonmomentum and measurement of the forward-backwardasymmetry are used. Electroweak radiative corrections usedfor the extraction of sin2 θlepteff and sin2 θW are derived from anapproach used at LEP-1 and SLD.Section II provides an overview of the lepton angular

distributions and the extraction of sin2 θlepteff . Section IIIdiscusses quantum chromodynamics (QCD) calculationsfor the forward-backward asymmetry and the inclusion ofelectroweak radiative-correction form factors used in theanalysis of high energy eþe− collisions. These form factorsare important in determining sin2 θW from the measurementof sin2 θlepteff . Section IV describes the experimental appa-ratus. Section V reports on the selection of data for themeasurement of the forward-backward asymmetry.Section VI describes the simulation of the reconstructeddata. Section VII presents the measurement of the asym-metry and the corrections made to the data and simulation.Section VIII describes the method used to extract sin2 θlepteff .Section IX describes the systematic uncertainties. Finally,Sec. X gives the results, and Sec. XI presents the summary.The units ℏ ¼ c ¼ 1 are used for equations and symbols,but standard units are used for numerical values of particlemasses and momenta, e.g., 40 GeV=c2 and 20 GeV=c,respectively, where c denotes the speed of light.

II. LEPTON ANGULAR DISTRIBUTIONS

The angular distribution of leptons from the Drell-Yanprocess in the rest frame of the boson is governed by the


072005-4

http://dx.doi.org/10.1103/PhysRevD.89.072005




polarization state of the γ�=Z boson. In amplitudes at ahigher order than the tree level, initial-state QCD inter-actions of the colliding partons impart transverse momen-tum, relative to the collision axis, to the γ�=Z boson. Thisaffects the polarization states.In the laboratory frame, the pp collision axis is the z

axis, with the positive z axis oriented along the direction ofthe proton. The transverse component of any vector, such asthe momentum vector, is defined to be relative to thez axis. The transverse component of vectors in otherreference frames is defined to be relative to the z axis inthose frames.The polar and azimuthal angles of the l− 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 thedirection of the incoming quark so that the definition of ϑparallels the definition used in eþe− collisions at LEP [7].This frame is approximated by the Collins-Soper (CS) restframe [10] for pp collisions. The rest frame is reached fromthe laboratory frame via two Lorentz boosts, first along thelaboratory z axis into a frame where the z component of thelepton-pair momentum vector is zero, followed by a boostalong the transverse component of the lepton-pair momen-tum vector. Within the CS frame, the z axis for the polarangle is the angular bisector between the proton directionand the reverse of the antiproton direction. The positive xaxis for the azimuthal angle is along the direction of thetransverse boost. Aview of the CS frame is shown in Fig. 1.By construction, the CS-frame angles ϑ and φ are invariantwith respect to boosts along the pp collision axis. When thetransverse momentum of the lepton pair is zero, the CS andlaboratory coordinate-system axes are the same, and the zaxis and quark directions coincide if the incoming quark ofthe Drell-Yan parton amplitude is from the proton.

The general structure of the Drell-Yan lepton angulardistribution in the boson rest frame consists of nine helicitycross-section ratios [11]:

dNdΩ

∝ ð1þ cos2ϑÞ þ A0

1

2ð1 − 3cos2ϑÞ þ A1 sin 2ϑ cosφ

þ A2

1

2sin2ϑ cos 2φþ A3 sin ϑ cosφþ A4 cos ϑ

þ A5sin2ϑ sin 2φþ A6 sin 2ϑ sinφþ A7 sinϑ sinφ:

(1)

The A0−7 coefficients are the ratios of the helicity crosssections for boson production relative to unpolarizedproduction, and are functions of kinematic variables ofthe boson. They vanish when the lepton-pair transversemomentum is zero, except for A4, which is present at thetree level of QCD and generates the forward-backward l−

asymmetry in cosϑ. Thus, at zero transverse momentum,the angular distribution reduces to the tree-level form1þ cos2 ϑþ A4 cos ϑ. The A4 coefficient is relatively uni-form across the range of transverse momentum where thecross section is large (under ∼45 GeV=c), but slowly dropsfor larger values of transverse momentum where the crosssection is very small. The A5−7 coefficients appear at thesecond order in the QCD strong coupling, αs, and are smallin the CS frame [11]. Hereafter, the angles ðϑ;φÞ and theangular coefficients A0−7 are specific to the CS rest frame.The A4 cos ϑ term is parity violating, and is due to the

interference of the amplitudes of the vector and axial-vectorcurrents. Its presence induces an asymmetry to theφ-integrated cos ϑ cross section. Two sources contribute:the interference between the Z-boson vector and axial-vector amplitudes, and the interference between the photonvector and Z-boson axial-vector amplitudes. The asym-metric component from the γ-Z interference cross section isproportional to gfA. The asymmetric component fromZ-boson self interference is proportional to a product ofgfV from the lepton and quark vertices, and thus is related tosin2 θW . At the Born level, this product is

Tl3ð1 − 4jQljsin2θWÞTq

3ð1 − 4jQqjsin2θWÞ;

where l and q denote the lepton and quark, respectively.For the Drell-Yan process, the quarks are predominantly thelight quarks: u, d, or s. The coupling factor has an enhancedsensitivity to sin2 θW at the lepton-Z vertex: assin2 θW ≈ 0.223, a 1% variation in sin2 θW changes thelepton factor (containing Ql) by about 8%, and it changesthe quark factor (containing Qq) by about 1.5% (0.4%) forthe u (d or s) quark. Electroweak radiative corrections donot significantly alter this Born-level interpretation. Loopand vertex electroweak radiative corrections are multipli-cative form-factor corrections to the couplings that changetheir value by a few percent.

lab+z

TP

APBP

BP--

+z

+x

FIG. 1 (color online). CS coordinate axes ðx; zÞ 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 within the rest frame. Relative to the

laboratory z axis, the transverse component of −ð~PA þ ~PBÞ isthe same as the transverse-momentum vector of the lepton pair in

the laboratory (~PT).


072005-5

For the description of the Drell-Yan process, therapidity, transverse momentum, and mass of a particleare denoted as y, PT, and M, respectively. The energy andmomentum of particles are denoted as E and P, respec-tively. In a given coordinate frame, the rapidity isy ¼ 1

2ln½ðEþ PzÞ=ðE − PzÞ�, where Pz is the component

of the momentum vector along the z axis of thecoordinate frame.The l− forward-backward asymmetry in cosϑ is

defined as

AfbðMÞ ¼ σþðMÞ − σ−ðMÞσþðMÞ þ σ−ðMÞ ¼

3

8A4ðMÞ; (2)

where M is the lepton-pair invariant mass, σþ is the totalcross section for cosϑ ≥ 0, and σ− is the total cross sectionfor cosϑ < 0. The sin2 θlepteff parameter is derived from theexperimental measurement of AfbðMÞ and predictions ofAfbðMÞ for various input values of sin2 θW . From theprediction that best describes the measured value ofAfbðMÞ, the value of sin2 θlepteff is derived. Electroweakand QCD radiative corrections are included in the pre-dictions of AfbðMÞ. The QCD predictions for AfbðMÞinclude electroweak radiative corrections derived from anapproach adopted at LEP [12].

III. ENHANCED QCD PREDICTIONS

Drell-Yan process calculations with QCD radiation donot typically include the full electroweak radiative correc-tions. However, the QCD, quantum electrodynamic, andweak corrections can be organized to be individually gaugeinvariant so that they can be applied separately andindependently.Quantum electrodynamic (QED) radiative corrections

which induce photons in the final state are not included inthe calculation of Afb. Instead, they are included in thephysics and detector simulation of the Drell-Yan processused in the measurement of Afb. For the processqq → lþl−, QED final-state radiation is most importantand is included in the simulation. The effects of QEDradiation are removed from the measured Afb.The Drell-Yan process and the production of quark pairs

in high-energy eþe− collisions are analog processes: qq →eþe− and eþe− → qq. At the Born level, the processamplitudes are of the same form except for the interchangeof the electrons and quarks. Electroweak radiative correc-tions, calculated and extensively used for precision fits ofLEP-1 and SLD measurements to the standard model [7],can be applied to the Drell-Yan process.In the remainder of this section, the technique used to

incorporate independently calculated electroweak radiativecorrections for eþe− collisions into existing QCD calcu-lations for the Drell-Yan process is presented.

A. Electroweak radiative corrections

The effects of virtual electroweak radiative correctionsare incorporated into Drell-Yan QCD calculations via formfactors for fermion-pair production in eþe− collisions,eþe− → Z → ff. The Z-amplitude form factors are calcu-lated by ZFITTER 6.43 [12], which is used with LEP-1 andSLD measurement inputs for precision tests of the standardmodel [7]. It is a semianalytical calculation for fermion-pairproduction and radiative corrections for high-energy eþe−collisions. Corrections to fermion-pair production via thevirtual photon include weak-interaction W-boson loops inthe photon propagator and Z propagators at fermion-photonvertices; these corrections are not gauge invariant exceptwhen combined with their gauge counterparts in the Zamplitude. The ZFITTER weak and QED corrections areorganized to be separately gauge invariant. Consequently,weak corrections to fermion-pair production via the virtualphoton are included with the Z-amplitude form factors. Therenormalization scheme used by ZFITTER is the on-shellscheme [13], where particle masses are on shell, and

sin2θW ¼ 1 −M2W=M

2Z (3)

holds to all orders of perturbation theory by definition.Since the Z-boson mass is accurately known (to�0.0021 GeV=c2 [7]), the inference of sin2 θW is equiv-alent to an indirect W-boson mass measurement.Form factors calculated by ZFITTER are stored for later

use in QCD calculations. The specific standard modelassumptions and parameters used in the form-factor cal-culation are presented in the Appendix. The calculatedform factors are ρeq, κe, κq, and κeq, where the label edenotes an electron and q denotes a quark. As thecalculations use the massless-fermion approximation, theform factors only depend on the charge and weak isospin ofthe fermions. Consequently, the stored form factors aredistinguished by three labels: e (electron type), u (up-quarktype), and d (down-quark type). The form factors arecomplex valued, and are functions of the sin2 θW parameterand the Mandelstam s variable of the eþe− → Z → ffprocess. The first three form factors of the amplitude areimportant. They can be reformulated as corrections to theBorn-level gfA and gfV couplings:

gfV →ffiffiffiffiffiffiffiρeq

p ðTf3 − 2Qfκfsin2θWÞ and gfA →

ffiffiffiffiffiffiffiρeq

pTf3 ;

where f ¼ e or q.The combination κf sin2 θW , called an effective-mixing

parameter, is directly accessible from measurements of theasymmetry in the cos ϑ distribution. However, neither thesin2 θW parameter nor the form factors can be inferred fromexperimental measurements without assuming the standardmodel. The effective-mixing parameters are denoted assin2 θeff to distinguish them from the on-shell definition ofsin2 θW [Eq. (3)]. The Drell-Yan process is most sensitive to


072005-6

the parameter sin2 θeff of the lepton vertex, κe sin2 θW ,which is commonly denoted as sin2 θlepteff . At the Z pole, κe isindependent of the quark flavor. For comparisons withother measurements, the value of sin2 θlepteff at the Z pole istaken to be ReκeðsZÞ sin2 θW (sZ ¼ M2

Z).

B. QCD calculations

The Drell-Yan QCD calculations are improved byincorporating the form factors from ZFITTER into theprocess amplitude. This provides an enhanced Bornapproximation (EBA) to the electroweak terms of theamplitude. The QED photon self-energy correction isincluded as part of the EBA. The photon amplitudeinfluences the shape of Afb away from the Z pole via itsinterference with the axial-vector part of the Z amplitude.The γ-Z interference, whose cross section is proportional toðs −M2

ZÞ, begins to dominate the total interference crosssection away from the Z pole. As the γ-Z interferencedilutes measurements of sin2 θeff , photonic corrections arealso included.The ZFITTER form factors ρeq, κe, and κq are inserted

into the Born gfA and gfV couplings for the Drell-Yanprocess. The κeq form factor is incorporated as an ampli-tude correction. Complex-valued form factors are used inthe amplitude. Operationally, only the electroweak-coupling factors in the QCD cross sections are affected.The standard LEP Z-boson resonant line shape and the totaldecay width calculated by ZFITTER are used.A leading-order (LO) QCD or tree-level calculation of

Afb for the process pp → γ�=Z → lþl− is used as thebaseline EBA calculation with ZFITTER form factors. It isused to provide a reference for the sensitivity of Afb to QCDradiation. The CT10 [14] next-to-leading-order (NLO)parton distribution functions (PDF) provide the incomingparton flux used in all QCD calculations discussed in thissection except where specified otherwise.Two NLO calculations, RESBOS [15] and the POWHEG-

BOX framework [16], are modified to be EBA-based QCDcalculations. For both calculations, the boson P2

T distribu-tion is finite as P2

T vanishes. The RESBOS calculationcombines a NLO fixed-order calculation at high bosonPT with the Collins-Soper-Sterman resummation formalism[17] at low boson PT, which is an all-orders summation oflarge terms from gluon emission. The RESBOS calculationuses CTEQ6.6 [18] NLO PDFs. The POWHEG-BOX is a fullyunweighted partonic-event generator that implementsDrell-Yan production of lþl− pairs at NLO. The NLOproduction implements a Sudakov form factor [19] thatcontrols the infrared diverence at low PT, and is constructedto be interfaced with parton showering to avoid doublecounting. The PYTHIA 6.41 [20] parton-showering algo-rithm is used to produce the final hadron-level event.The RESBOS and POWHEG-BOX NLO calculations are

similar and consistent. The RESBOS calculation is chosen as

the default EBA-based QCD calculation of Afb with variousinput values of sin2 θW . As the POWHEG-BOX NLO programhas a diverse and useful set of calculation options, it is usedto estimate QCD systematic uncertainties.

IV. THE EXPERIMENTAL APPARATUS

The CDF II apparatus is a general-purpose detector [21]at the Fermilab Tevatron pp collider whose center-of-momentum (cm) energy is 1.96 TeV. The positive z-axis isdirected along the proton direction. For particle trajectories,the polar angle θcm is relative to the proton direction and theazimuthal angle ϕcm is oriented about the beam line axiswith π=2 being vertically upwards. The component ofthe particle momentum transverse to the beam line isPT ¼ P sin θcm. The pseudorapidity of a particle trajectoryis η ¼ − ln tanðθcm=2Þ. Detector coordinates are specifiedas ðηdet;ϕcmÞ, where ηdet is the pseudorapidity relative tothe detector center (z ¼ 0).The central charged-particle tracking detector (tracker) is

a 3.1 m long, open-cell drift chamber [22] that extendsradially from 0.4 to 1.4 m. Between the Tevatron beam pipeand the central tracker is a 2 m long silicon tracker [23].Both trackers are immersed in a 1.4 T axial magnetic field.Outside the drift chamber is a central barrel calorimeter[24,25] that covers the region jηdetj < 1.1. The forwardend-cap regions are covered by the end-plug calorimeters[26–28] that cover the regions 1.1 < jηdetj < 3.5.The muon detectors are outer charged-particle trackers

that are positioned behind iron hadron absorbers. Theprimary absorbers are the calorimeters. There are fourseparate detectors, denoted CMU, CMP, CMX, and BMU.The CMU muon detector [29], located just beyond thecentral barrel calorimeter, has a cylindrical geometry andcovers the region jηdetj < 0.6. The central calorimeterprovides approximately 5.5 pion (4.6 nuclear) interactionlengths of shielding. The CMP muon detector shadows theCMU detector, covers the same region, jηdetj < 0.6, but hasa rectangular geometry. There are an additional 2.3 pioninteraction lengths of shielding between the CMP andCMU detectors. The CMX muon detectors cover theregions 0.6 < jηdetj < 1.0, and are located behindapproximately 6.2 pion interaction lengths of shielding.The BMU muon detectors cover the forward regions1.0 < jηdetj < 1.5, and are situated behind at least 6.2 pioninteraction lengths of shielding.

V. DATA SELECTION

The data set, collected over 2002–2011, is the full CDFRun II data set and consists of pp collisions correspondingto an integrated luminosity of 9.2 fb−1. Section VA reportson the online selection of events (triggers) for the Afbmeasurement. Section V B describes the offline selection ofmuon candidates, and Sec. V C describes the selection ofmuon pairs.


072005-7

A. Triggers

Muon candidates used in this analysis are selected fromtwo online triggers: CMUP_18 and CMX_18 [30–33]. Theseselections require at least one muon candidate in the eventto be in the region jηdetj < 1. The CMUP_18 selectionaccepts muon candidates based on track segments recon-structed in the CMU and CMP detectors that are geomet-rically matched to a PT > 18 GeV=c charged-particletrack. The CMX_18 selection accepts muon candidates witha PT > 18 GeV=c charged-particle track in the centraltracker that is matched to a track in the CMX muondetector.

B. Offline muon selection

The offline selection begins with a charged-particle trackcandidate in the central tracker. The track is extrapolatedthrough the calorimeters and into the muon detectors forassociation with independent track segments reconstructedin the muon detectors. The selection is based on the qualityof track-to-segment matching and energy deposition in thecalorimeters. The energy deposition in the calorimetersmust be consistent with that of a minimum-ionizingparticle. The track-to-segment matching is applied onlyif the track extrapolates into a fiducial region of a muondetector. The selection criteria used [21] are stringent andresult in a well reconstructed sample of muon candidateswith high purity.The categories of muon candidates with associated

segments in a muon detector are denoted with the followinglabels: CMUP, CMU, CMP, CMX, and BMU. For theCMUP category, the track extrapolation has matchingsegments in both the CMU and CMP detectors. TheCMU category comprises muons with a matching segmentin the CMU detector only. The CMP category comprisesmuons with a matching segment in the CMP detector only.The muons in the CMX and BMU categories havematching segments in the CMX and BMU muon detectors,respectively.As the coverage of the muon detectors has gaps, muon

candidates without associated segments in a muon detectorare also used. They consist of tracks that extrapolate intononfiducial regions of the muon detector, and fiducialtracks without matching segments. This category is denotedas CMIO (minimum-ionizing category), and consists ofmuon candidates that satisfy the track-quality andminimum-ionization energy loss requirement in thecalorimeters.The acceptance for muon candidates is limited by the

geometric acceptance of the central tracker, whose accep-tance of tracks is uniform up to jηj ≈ 1.1 but then fallsrapidly and vanishes at jηj ≈ 1.5. In the jηj > 1.1 region, thetrack quality requirements for muons in the BMU categoryare relaxed. However, the track-quality requirements forCMIO muons that have no associated muon detectorsegments are kept stringent.

C. Offline muon-pair selection

Events are required to contain two muon candidates. Thekinematic and fiducial acceptance region for muons andmuon pairs used in the Afb measurement are listed below.(1) Muon kinematics and fiducial criteria

(a) PT > 20 GeV=c;(b) Muon 1: CMUP or CMX category;(c) Muon 2: any muon category.

(2) Muon-pair criteria(a) Muon 1 and 2: oppositely charged;(b) jyj < 1;(c) Muon-pair mass M > 40 GeV=c2.

One of the muons, denoted by “Muon 1,” is a CMUP orCMX muon that is consistent with the online selection. Asthe second muon can belong to any one of the six muoncategories, eleven muon-pair topologies are possible. Muonpairs consistent with the passage of cosmic rays through thedetector are rejected [21]. The limited acceptance of thecentral tracker restricts the accepted rapidities (y) ofthe muon pairs. As there is limited acceptance forjyj > 1, the Afb measurement is restricted to muon pairsin the kinematic region of jyj < 1.The number of events passing all requirements, after

background subtraction, is 276 623. The fraction of eventsin each of the various muon-pair topologies is summarizedin Table I. As the two topologies with CMP muons are rare,they are combined. The backgrounds are from QCD and theelectroweak (EWK) processes of WW, WZ, ZZ, tt,W þ jets, and Z → τþτ−. The QCD background is pri-marily from dijets in which a particle in a jet has penetratedthe shielding. The high-PT muon sources yield at least onereal muon. The second muon is either a real second muonor a track that is misidentified as a muon.The EWK-process backgrounds are derived from PYTHIA

[34] samples that are processed with the CDF II detectorsimulation, and in which the integrated luminosity of eachsample is normalized to the data. TheWW,WZ, ZZ, and ttsamples are NLO simulations. As the W þ jets and Z →τþτ− processes are LO simulations, the total cross section

TABLE I. Fraction of events after background subtraction forthe various muon-pair topologies. The total number of events is276 623.

Muon 1 Muon 2 Fraction

CMUP CMUP 0.159CMUP CMX 0.252CMUP CMU 0.067CMUP CMIO 0.181CMUP BMU 0.057CMX CMX 0.095CMX CMU 0.052CMX CMIO 0.111CMX BMU 0.025CMUPþ CMX CMP 0.002


072005-8

used for the calculation of the integrated luminosityincludes a NLO-to-LO K-factor of 1.4. The EWK-background events that pass the selection criteria amountto 0.53% of the total sample.The QCD backgrounds are estimated from the data with

the number of same-charge muon pairs in the sample, andamount to 0.10% of the total sample. The muon-pairinvariant mass distributions for the data and the back-grounds are shown in Fig. 2. The distribution of same-charge muon pairs from the data sample shown in Fig. 2also provides a measure of muon-charge misidentification.Events in which Z → μþμ− decays are incorrectly recon-structed as same-charge muon pairs form a Z-resonancepeak within the same-charge mass distribution. From thedistribution of same-charge muon pairs shown in Fig. 2, itis concluded that charge misdentification is negligible.Backgrounds are subtracted in the measurement of Afb,

and the method is presented in Sec. VII A.

VI. SIGNAL SIMULATION

Drell-Yan pair production is simulated using theMonte Carlo event generator, PYTHIA [34], and CDF IIdetector-simulation programs. PYTHIA generates the hard,leading-order QCD interaction, qþ q → γ�=Z, simulatesinitial-state QCD radiation via its parton-shower algo-rithms, and generates the decay γ�=Z → lþl−. TheCTEQ5L [35] nucleon parton distribution functions areused in the QCD calculations. The underlying-event andboson-PT parameters are derived from the PYTHIA con-figuration AW (i.e., PYTUNE 101, which is a tuning toprevious CDF data) [34,36,37]. The generator-level PTdistribution is further adjusted so that the shape of thereconstruction-level, simulated PT distribution matchesthe data.Generated events are processed by the CDF II event and

detector simulation. The detector simulation is based onGEANT-3 and GFLASH [38]. The event simulation includes

PHOTOS 2.0 [39,40], which adds final-state QED radiation(FSR) to decay vertices with charged particles (e.g.,γ�=Z → μμ). The default implementation of PYTHIA plusPHOTOS (PYTHIA+PHOTOS) QED radiation in the simulationinfrastructure has been validated in a previous measurementof sin2 θlepteff using Drell-Yan electron pairs [5].The time-dependent beam and detector conditions are

modeled appropriately in the simulation, including the pand p beam line parameters; the luminous region profile;the instantaneous and integrated luminosities per data-taking period; and detector component calibrations, whichinclude channel gains and malfunctions. The simulatedevents are reconstructed, selected, and analyzed in the sameway as the experimental data.

VII. THE Afb MEASUREMENT

The Collins-Soper frame angle, cos ϑ [10], is recon-structed using the following laboratory-frame quantities:the lepton energies, the lepton momenta along the beamline, the dilepton invariant mass, and the dilepton transversemomentum. The angle of the negatively charged lepton is

cosϑ ¼ l−þlþ− − l−−lþþ

MffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiM2 þ P2

T

p ;

where l� ¼ ðE� PzÞ and the þ ð−Þ superscript specifiesthat l� is for the positively (negatively) charged lepton.Similarly, the Collins-Soper expression for φ in terms oflaboratory-frame quantities is

tanφ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiM2 þ P2

T

p

M

~Δ · RT

~Δ · PT

;

where ~Δ is the difference between the l− and lþ mo-mentum vectors; RT is the transverse unit vector along~Pp × ~P, with ~Pp being the proton momentum vector and ~Pthe lepton-pair momentum vector; and PT is the unit vectoralong the transverse component of the lepton-pair momen-tum vector. At PT ¼ 0, the angular distribution is azimu-thally symmetric.The Afb is measured in 16 mass bins, starting with

M ¼ 50 GeV=c2. This section details the measurementmethod, which includes corrections to the data and thesimulation, and presents the fully corrected measurement.The key components of the measurement are introduced inthe next two sections: Sec. VII A describes a newlydeveloped event-weighting technique, and Sec. VII Bdescribes the muon momentum and resolution calibration.Section VII C describes the data-driven corrections appliedto the simulated data. Section VII D describes the reso-lution-unfolding technique and the corresponding covari-ance matrix of the unfolded Afb measurement. Section VIIE describes the final corrections to the measurement andpresents the fully corrected measurement of Afb.

) 2M (GeV/c40 50 60 70 80 90 100 110 120 130 140

2E

vent

s / 0

.25

GeV

/c

-110

1

10

210

310

410

FIG. 2 (color online). Muon-pair invariant mass distributions.The upper set of crosses is the background-subtracted data, themiddle set of crosses is the EWK background, and the lower setof crosses is the QCD background (same-charge muon pairs). TheEWK background is derived from simulation.


072005-9

A. Event-weighting method

The forward-backward asymmetry Afb of Eq. (2) istypically determined in terms of the measured cross sectionσ ¼ N=ðLεAÞ, where N is the number of observed eventsafter background subtraction, L is the integrated luminos-ity, ϵ is the reconstruction efficiency, and A is theacceptance within the kinematic and fiducial restrictions.The expression is

Afb ¼Nþ=ðϵAÞþ − N−=ðϵAÞ−Nþ=ðϵAÞþ þ N−=ðϵAÞ− :

The terms Nþð−Þ and ðϵAÞþð−Þ, respectively, represent Nand ϵA for candidates with cosϑ ≥ 0 (cosϑ < 0). Eachmuon-pair topology listed in Table I requires a separateevaluation of ðϵAÞ�.The Afb is measured using a new and simpler technique:

the event-weighting method [41]. The method is equivalentto measurements of Afb in j cosϑj bins with these simplify-ing assumptions:(1) ðϵAÞþ ¼ ðϵAÞ− in each j cos ϑj bin, and(2) Eq. (1) describes the angular distributions.

The measurement of Afb within a j cosϑj bin (A0fb) only

depends on N�, but is related to Afb through an angulardependence,

A0fb ¼

Nþ − N−

Nþ þ N− ∝ Afbj cosϑj

1þ cos2ϑþ � � � ; (4)

where 1þ cos2ϑþ � � � denotes symmetric terms in Eq. (1).The j cosϑj term arises from the difference in the numeratorNþ − N−, and the 1þ cos2 ϑþ � � � term arises from thesum in the denominator Nþ þ N−. As the angular factor isthe equivalent of an importance-sampling function ofMonte Carlo simulations, the binned measurements arereformulated into an unbinned, event-by-event weightedexpression,

Afb ¼Nþ

n − N−n

Nþd þ N−

d: (5)

TheN�n andN�

d terms represent weighted event counts, andthe subscripts n and d signify the numerator and denom-inator sums, respectively, which contain the same eventsbut with different event weights. The weights take intoaccount the angular terms of the numerator and denomi-nator sums, and include a statistical factor for the expectedmeasurement uncertainty at each value of j cosϑj, theinverse of the square of the angular factor in A0

fb.Consequently, the method is equivalent to using a maxi-mum-likelihood technique, and for an ideal detector, thestatistical precision of Afb is expected to be about 20%better relative to the direct counting method [41]. However,detector resolution and limited acceptance degrade theideal gain.

The event weights are functions of the reconstructedkinematic variables, cosϑ, φ, and the muon-pair variables,M and PT. Only the A0 and A2 terms of Eq. (1) are used inthe denominator of the angular factor of Eq. (4), and theangular coefficients are parametrized with

A0 ¼ A2 ¼kP2

T

kP2T þM2

;

where k is a tuning factor for the PT dependence of the A0

and A2 coefficients. For this analysis, k ¼ 1.65, which isderived from a previous measurement of angular coeffi-cients [42]. The exact form of these angular terms in theevent weights has very little impact on Afb because the bulkof the events is at low boson PT. The difference betweenk ¼ 1 and k ¼ 1.65 is negligible.The EWK and QCD backgrounds are subtracted from

the weighted event sums on an event-by-event basis.For the QCD same-charge pair background, cos ϑ iscalculated by randomly assigning a lepton of each pairas the negatively charged lepton. Background eventspassing the selection requirements are assigned negativeevent weights when combined with the event sums.The event-weighting method does not compensate the

following sources of bias:(1) smearing of kinematic variables due to the detector

resolution,(2) kinematic regions with limited acceptance, and(3) detector nonuniformity resulting in ðϵAÞþ ≠ ðϵAÞ−.

Resolution-smearing effects are unfolded with the aid ofthe simulation. For the unfolding to be accurate, the muonmomentum scale and resolution for both the data andsimulation are precisely calibrated. In addition, the cosϑand muon-pair invariant-mass distributions of thesimulation are matched to agree with the data.After resolution unfolding, the event-weighted Afb can

have a small, second-order bias. The bias is estimatedusing the simulation and is the difference between the truevalue of Afb from the physics events generated by PYTHIA

and the result of the measurement on the simulatedsample. One source of bias is from the limited muon-pair acceptance at forward rapidities. There is a smallincrease in the forward-backward physics asymmetrywith increasing jyj for jyj ≈ 1 and above. As theevent-weighted Afb is an average of Afb over the ydistribution of accepted events, regions with significantlylimited or no acceptance bias the average. The kinematicrestriction of jyj < 1 specified in Sec. V C reduces thisbias. Another possible source of bias is from detectornonuniformity: ðεAÞþ ≠ ðεAÞ−. This distorts the estimateof A0

fb [Eq. (4)]. The effects of these biases, which arequantified later in Sec. VII E, are removed from the Afbmeasurement.


072005-10

B. Muon momentum calibration

The typical dependence of Afb as a function of thelepton-pair invariant mass is shown in Fig. 3. Withmomentum miscalibrations, an event produced at massM with asymmetry AfbðMÞ is associated with a differentmass M0. The measured AfbðM0Þ becomes biased becauseof this systematic dilution. The correct calibration of themuon momentum is critical for the measurementof AfbðMÞ.The momentum calibration procedure is adapted from a

technique developed for the Compact Muon Spectrometer(CMS) [43]. The general principles are briefly describednext, followed by the CDF adaptation. The tracker is splitinto regions of ðη;ϕÞ. For each region, track curvaturecorrections are determined. They are the curvature scalecorrection to the magnetic-field path integral

RB · dl and

the tracking alignment offset, which are denoted by 1þ sand o, respectively. The corrections s and o are the same forpositively and negatively charged particles. For an inputtrack curvature C, the corrected curvature is ð1þ sÞCþ o.In the following discussion, the curvature C is synonymousto the charge-signed 1=PT of a track.The calibration sample consists of oppositely charged

muon pairs enriched in Z-boson decays. The muons in thesample are binned according to their ðη;ϕÞ trajectories. Thecharge-signed 1=PT for the μ� is denoted by C�, and itsdistribution in each bin has a sharp peak if the muon pairsare produced in the center of the tracker. The peaks becomenarrower as the Z-boson mass selection window is madesmaller. The calibration method requires a single distinctpeak in the C� distributions. The locations of these peaksare calibrated against simulated Drell-Yan muon-pairevents that pass the calibration sample selection criteria.The calibration ansatz is that the 1þ s and o parametersmap the peaks for C� onto the true positions predicted bythe simulation. The true location of the peaks (the truth) is

the generator-level charge-signed 1=PT of the μ� after QEDFSR, and it is denoted by C�

true. Thus, the calibrationconstraints for s and o are given by

Cþtrue ¼ ð1þ sÞCþ þ o C−

true ¼ ð1þ sÞC− þ o:

For the CDF calibration, muon pairs in the Z-bosonregion of 76 < M < 106 GeV=c2 are used. There are 262000 events in the sample, with very little background. Themuons are binned using their ðη;ϕÞ trajectories: eight fixed-width ϕ bins and eight variable-width η bins. The η binsspan the range of −1.6 to 1.6, with bin boundaries of −1.6,−1.0, −0.6, −0.3, 0.0, 0.3, 0.6, 1.0, and 1.6. These bins arefurther divided into SS and opposite-side (OS) muon-pairtopologies: SS pairs have η1η2 ≥ 0 and OS pairs haveη1η2 < 0, where the subscript 1 (2) denotes Muon 1 (2).The peak of the curvature spectrum for OS-pair muons ismore dependent on their point of origin along the z axisthan for SS-pair muons. At the Tevatron, the broadluminous region of pp collisions (30 cm longitudinalrms) has a significant impact on the higher jηj bins.Figures 4 and 5 show the Cþ distribution for SS and OSpairs in one bin of the central η region of ð−0.3; 0Þ. Thegenerator-level C� distributions provide an adequatedescription of the data for the initial steps of the iterativecalibration procedure.The momentum scale calibration is iterative because the

s and o calibration parameters affect the shape and locationof the peaks. For the high η bins, the calibration accuracy isno better than 1% due to the limited number of calibrationevents. After the third iteration with curvature peaks, thesharper Z-boson peak in the muon-pair invariant massdistribution is used to determine the s and o calibrationparameters. The final three iterations use the mass peaks.For the calibration using the muon-pair invariant mass, onemuon is selected as the tagwhich determines the bin. Thereis no bin restriction on the second muon.

)2M (GeV/c

40 60 80 100 120 140 160 180 200

fbA

-0.4

-0.2

0

0.2

0.4

0.6

FIG. 3. Typical dependence of Afb as a function of the lepton-pair invariant mass. The curve is an analytic calculation. Thevertical line is at M ¼ MZ.

-1 (GeV/c)T1/P

0 0.01 0.02 0.03 0.04 0.05

-1E

vent

s / 0

.001

(G

eV/c

)

0

50

100

150

200

250

FIG. 4 (color online). Distribution of Cþ for same-side (SS)pairs in the central η region of ð−0.3; 0Þ. The crosses are the data,and the solid histogram is the generator-level distributionnormalized to the data.


072005-11

The momentum scale calibration is applied to both thedata and simulation. Bins that are perfectly calibrated havecorrection values s ¼ 0 and o ¼ 0. The distribution ofcorrections for the data is much wider than that for thesimulation. In addition, corrections for the high jηj bins arewider than those for the central region bins. For thedata, the mean scale correction s from the 128 calibrationbins is 0.1%, and the mean alignment offset o is−0.02 ðTeV=cÞ−1. The rms of the scale corrections is0.4%, and the rms for the alignment offset correctionsis 0.3 ðTeV=cÞ−1, or 1.4% at PT ¼ MZ=2. For the simu-lation, the mean scale correction and the mean alignmentoffset are 0.1% and −0.01 ðTeV=cÞ−1, respectively, and thecorresponding rms values are 0.08% and 0.03 ðTeV=cÞ−1,respectively. The calibration of both the data and simulationsets their absolute momentum scales to the generator-levelCtrue scale after QED FSR.The momentum resolution for the simulation is cali-

brated to the momentum resolution of the data after thescale calibrations. The resolution calibration uses the initialcurvature of the simulated data, C. The bias of thiscurvature relative to its true value for each event is

ΔCtrue ¼ Ctrue − C:

The resolution is modified by changing the amount of biason an event-by-event basis with the parameter f,

C0 ¼ C − fΔCtrue;

where C0 is the new curvature. Relative to the original Cdistribution, the rms of the C0 distribution is changed by thefactor 1þ f. The mass distributions of muon pairs in the86–96 GeV=c2 region of the data and simulation are usedto determine f. The value that provides the best match tothe data is f ¼ þ0.15, and the χ2 of the simulation-to-datacomparison is 68 over 79 bins.

The momentum scale and resolution calibrations dependon the agreement between the simulated and experimental-data distributions for the PT of the muons and invariantmass of the pair. The full results of the momentum scale andresolution calibration are presented in the next section,which describes the data-driven corrections to thesimulation.

C. Corrections to the simulation

The simulation presented in Sec. VI does not describethe data accurately enough for the Afb measurement.Additional corrections applied to the simulated data aredescribed in this section. All corrections are scale factors,or event weights, that are applied to simulated events. Boththe simulated and experimental data are divided into thesame 39 time periods used for the offline calibration ofCDF data.The first set of corrections are event-wide corrections.

The event selections described in Sec. V are applied toboth the simulated and experimental data. For each muon-pair topology (Sec. V C), the number of events is adjustedperiod by period to match the data. This adjustmentcontains corrections to the integrated luminosity, the triggerefficiency, and global reconstruction efficiencies for eachmuon-pair topology. The distributions of the number of ppcollision vertices in each event ðnvtxÞ and the location ofthese vertices along the beam line ðzvtxÞ changed signifi-cantly with improvements to the Tevatron collider. Thesedistributions are inadequately simulated. The nvtx distribu-tion is corrected on a period-by-period basis. The zvtxcorrection is split into a smaller set of seven correctionblocks.The momentum scale calibration described in the pre-

vious section is applied to both the simulated and exper-imental data. The momentum resolution of the simulateddata is then adjusted to match the resolution of theexperimental data. After these calibrations, the muon-pairinvariant mass distribution of the simulated data is in goodagreement with that of the experimental data. The massdistributions are shown in Figs. 6 and 7. The muon PTdistributions are shown in Figs. 8 and 9.As the Collins-Soper cos ϑ distribution is important for

corrections to the Afb measurement, the simulated cosϑdistribution is adjusted to improve agreement with the data.The adjustments, determined for eight muon-pair invariantmass bins whose boundaries are aligned with those used inthe measurement, are determined from the ratios of thedata-to-simulation cos ϑ distributions. The ratios are para-metrized with the function p0 þ p1 cosϑþ p2cos2ϑ, wherep0, p1, and p2 are fit parameters. In the fits of the ratioswith this function, the values of the asymmetry-differenceparameter p1 are consistent with zero. The ratios are welldescribed by the symmetric function with p1 ¼ 0, which isused for the adjustments. The parametrized ratios arenormalized to preserve the event count for the mass bin.

-1 (GeV/c)T1/P

0 0.01 0.02 0.03 0.04 0.05

-1E

vent

s / 0

.001

(G

eV/c

)

0

50

100

150

200

250

FIG. 5 (color online). Distribution of Cþ for OS pairs in thecentral η region of ð−0.3; 0Þ. The crosses are the data, and thesolid histogram is the generator-level distribution normalized tothe data.


072005-12

The adjustment for the bin containing the Z pole is uniformin cosϑ. In bins away from the Z pole, the adjustmentsredistribute events from the periphery of the cosϑ distri-bution to its center (cosϑ ≈ 0). With increasing distances ofthe mass bin from the Z pole, the fraction of redistributedevents increases, but remains under 5%. The cosϑ dis-tribution after the adjustments is shown in Fig. 10. Thedefault φ distribution is adequate and is shown in Fig. 11.

D. Resolution unfolding

After applying the calibrations and corrections to theexperimental and simulated data, the Afb is measured inbins of the muon-pair invariant mass with the event-weighting method. This measurement is denoted as theraw Afb measurement because the event-weighting methodprovides a first-order acceptance correction, but does notinclude resolution unfolding and final-state QED radiation.The raw Afb measurement is shown in Fig. 12.Resolution unfolding uses the event transfer matrices

from the simulation, denoted by ngr. This symbol identifiesthe number of selected events that is generated in the muon-pair ðM; cosϑÞ bin g and reconstructed in the ðM; cosϑÞ

)2M (GeV/c70 75 80 85 90 95 100 105 110 115

2E

vent

s / 0

.25

GeV

/c

0100020003000400050006000700080009000

10000

FIG. 6 (color online). Calibrated muon-pair invariant massdistributions. The crosses are the background-subtracted dataand the solid histogram is from the simulation. The comparison ofthe simulation with the data yields a χ2 of 219 for 200 bins.

)2M (GeV/c

40 50 60 70 80 90 100 110 120 130 140

2E

vent

s / 0

.25

GeV

/c

10

210

310

410

FIG. 7 (color online). Calibrated muon-pair invariant massdistributions over an extended mass range. The crosses are thebackground-subtracted data and the solid histogram is from thesimulation. The comparison of the simulation with the data yieldsa χ2 of 518 for 400 bins.

(GeV/c)T

Leading muon P20 30 40 50 60 70 80 90 100

2E

vent

s / 0

.5 G

eV/c

0

2000

4000

6000

8000

10000

FIG. 8 (color online). Calibrated PT distribution for the muonwith the larger PT. The crosses are the background-subtracteddata and the solid histogram is from the simulation.

(GeV/c)T

Sub-leading muon P20 30 40 50 60 70 80 90 100

2E

vent

s / 0

.5 G

eV/c

0

2000

4000

6000

8000

10000

FIG. 9 (color online). Calibrated PT distribution for the muonwith the smaller PT. The crosses are the background-subtracteddata and the solid histogram is from the simulation.

ϑcos-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

Eve

nts

/ 0.0

2

0500

100015002000250030003500400045005000

FIG. 10 (color online). Adjusted cos ϑ distribution in theCollins-Soper frame. The crosses are the background-subtracteddata and the solid histogram is from the simulation.


072005-13

bin r. Sixteen mass bins are defined. Their boundaries are50, 80, 82, 84, 86, 88, 89, 90, 91, 92, 93, 94, 96, 98, 100,102, and 1000 GeV=c2. The 50–80 and 102–1000 GeV=c2

bins are referenced as the underflow and overflow bins,respectively. The forward-backward asymmetry hastwo angular regions, cosϑ ≥ 0 ðþÞ and cosϑ < 0 ð−Þ.Operationally, 32 × 32 square transfer matrices for a 32-element state vector are implemented. The first 16 elementsof the vector are the mass bins for theþ angular region, andthe remaining 16 elements are for the − angular region.The simulation predicts significant bin-to-bin event

migration among the mass bins when the produced andreconstructed values of cosϑ have the same sign. For amass bin, there is very little migration of events from oneangular region to the other. As the simulation sample size isnormalized to the integrated luminosity of the data, thetransfer matrices provide properly normalized estimates ofevent migration between bins. An estimator for the true

unfolding matrix is Ugr ¼ ngr=Nr, where Nr ¼P

gngr isthe expected total number of weighted events reconstructedin bin r. The 32-element state vector for Nr is denoted as~Nr, and the matrix Ugr by U. The estimate for theresolution-unfolded state vector of produced events is~Ng ¼ U · ~Nr. The accuracy of the simulation of U isdetermined by the sample size of the data used forcalibrations and corrections.For the event-weighting method, there are two transfer

matrices that correspond to the weighted event counts Nnand Nd of Eq. (5), and thus two separate unfolding matrices

U and two separate event-weighted measurements of ~Nr.

They are used to estimate the two resolution-unfolded ~Ng

vectors from which Afb is derived. The measurements of Afb

for the 16 mass bins are collectively denoted by ~Afb.The covariance matrix of the Afb measurement, denoted

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 · ð ~Nr þ δ ~NrÞ, where

δ ~Nr represents a fluctuation from the expectation ~Nr.

The variation δ~Afb resulting from the ~Ng fluctuation isensemble averaged to obtain the covariance matrix

Vlm ¼ hðδ~AfbÞlðδ~AfbÞmi;

where ðδ~AfbÞk (k ¼ l and m) denotes the kth element of

δ~Afb. Each element i of ~Nr receives independent, normallydistributed fluctuations with a variance equal to the valueexpected for Ni. Because Ni is a sum of event weights,fluctuations of Ni are quantified with the variance of itsevent weights. The two ~Nr vectors, the numerator vectorand the denominator vector, have correlations. Elements iof the numerator and denominator vectors contain the sameevents, the only difference being that they have differentevent weights. To include this correlation, the event-countvariations of elements i of the numerator and denominatorδ ~Nr vectors are based on the same fluctuation from anormal distribution with unit rms.The covariance matrix is expanded and inverted to the

error matrix using singular-value decomposition (SVD)methods. As the covariance matrix is a real-valued sym-metric 16 × 16 matrix, its 16 eigenvalues and eigenvectorsare the rank-1 matrix components in the decomposition ofthe covariance matrix and the error matrix

V ¼X

n

λnð~vn~vnÞ and V−1 ¼X

n

λ−1n ð~vn~vnÞ;

where λn and ~vn are the eigenvalues and eigenvectors of V,respectively, and ð~vn~vnÞ represents a vector projectionoperator, i.e., jvnihvnj in the style of Dirac bra-kets.

ϕ-3 -2 -1 0 1 2 3

Eve

nts

/ 0.0

628

0

500

1000

1500

2000

2500

3000

3500

4000

FIG. 11 (color online). Observed φ distribution in the Collins-Soper frame. The crosses are the background-subtracted data andthe solid histogram is from the simulation.

)2M (GeV/c50 100 150 200 250 300

fbR

aw A

-0.4

-0.2

0

0.2

0.4

0.6

0.8

DataSimulation: PYTHIA+PHOTOSPrediction: PYTHIA |y| < 1

FIG. 12 (color online). Raw Afb measurement in bins of themuon-pair invariant mass. Only statistical uncertainties areshown. The PYTHIA jyj < 1 asymmetry curve does not includethe effect of QED FSR.


072005-14

The covariance matrix has several eigenvalues with verysmall values. They can be interpreted as simulation noise.While they contribute very little to the structure of thecovariance matrix, they completely dominate the errormatrix. Consequently, comparisons between the Afb meas-urement and predictions that use the error matrix areunstable. An SVD method to alleviate this instablility isused, and presented in Sec. VIII.

E. Event-weighting bias correction

After resolution unfolding, the event-weighted Afb val-ues can have second-order acceptance and reconstruction-efficiency biases. The most significant is the measurementbias from regions of limited boson acceptance, and to alesser extent, from detector nonuniformities resulting inðϵAÞþ ≠ ðϵAÞ−. The limited rapidity acceptance of muonpairs is shown in Fig. 13. As jyj increases, Afb slowlyincreases, and this increase is not fully taken into account inthe regions of limited boson acceptance.The bias is defined as the difference between the true

value of Afb calculated from the underlying events gen-erated by PYTHIA and the simulation estimate. The estimateis the value of the resolution-unfolded Afb obtained fromthe event-weighted simulation. Kinematic distributions ofthe simulated data that are important for the unfoldingmatrix are adjusted to agree with the data, but the adjust-ments exclude terms linear in the cosϑ kinematic variable.Linear adjustments can only be applied to the underlyingphysics distribution and propagated to the observed cosϑdistribution. The bias is a mass-bin by mass-bin additivecorrection to the unfolded Afb measurement, and is shownin Fig. 14. A small net positive bias is expected due to thelimited acceptance at the edges of the jyj < 1 measurementregion for muon pairs; a bias of ð0.0009� 0.0005Þ isobserved. The fully corrected measurement of Afb, includ-ing the bias correction, is shown in Fig. 15 and tabulated inTable II.

VIII. Extraction of sin2θlepteff

The EWK mixing parameters sin2 θlepteff and sin2 θW areextracted from the Afb measurement presented in Fig. 15using Afb templates calculated using different values ofsin2 θW . Three EBA-based calculations are used: LO (tree),RESBOS NLO, and POWHEG-BOX NLO. For the EBAelectroweak form-factor calculations, the weak-mixingparameter is sin2 θW .The Afb measurement is directly sensitive to the

effective-mixing parameters sin2 θeff , which are combina-tions of the form factors and sin2 θW (Sec. III A). The Drell-Yan Afb is most sensitive to the effective-leptonic sin2 θlepteff .While the extracted values of the effective-mixing param-eters are independent of the details of the EBA model, theinterpretation of the best-fit value of sin2 θW and its

y-1.5 -1 -0.5 0 0.5 1 1.5

Eve

nts

/ 0.0

2

0

500

1000

1500

2000

2500

3000

3500

4000

4500

FIG. 13 (color online). Muon-pair y distribution. The crossesare the background-subtracted data and the histogram is thesimulation. The measurement is resticted to the region jyj < 1.The upper curve is the (arbitrarily normalized) shape of theunderlying rapidity distribution from PYTHIA.

)2M (GeV/c80 85 90 95 100

fb e

stim

ated

A- -

fbT

rue

A

-0.02

-0.01

0

0.01

0.02First/Last Bin: Underflows/Overflows

FIG. 14 (color online). Event-weighting bias for each of themuon-pair invariant mass bins. The bias is estimated with thesimulation, and the uncertainties represent the full precision ofthe simulation.

)2M (GeV/c80 85 90 95 100

fbA

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5First/Last Bin: Underflows/Overflows

DataPYTHIARESBOS NLO EBA

FIG. 15 (color online). Fully corrected Afb. The measurementuncertainties are uncorrelated bin-by-bin unfolding estimates.The vertical line is M ¼ MZ. The PYTHIA calculation usessin2 θlepteff ¼ 0.232. The EBA-based RESBOS calculation uses

sin2 θW ¼ 0.2233 ðsin2 θlepteff ¼ 0.2315Þ.


072005-15

corresponding form factors are dependent on the details ofthe EBA model.The measurement and templates are compared using the

χ2 statistic evaluated with the Afb measurement errormatrix. A regularization term is added to the eigenvaluecoefficients of the SVD expansion of the error matrix toattenuate the contributions of noise terms with smalleigenvalues. The statistical uncertainties of the bias cor-rection and the template calculation are used as uncorre-lated regularization terms. Each uncertainty is projectedonto the eigenvector basis of the covariance matrix and thenapplied in quadrature as a regularization term:

λn → λn þX

i

ð~vnÞ2iΔ2i ;

whereΔi is the uncertainty for mass bin i, and λn and ~vn arethe eigenvalue and eigenvector, respectively, of the covari-ance matrix basis vector n. In the basis of the diagonalmeasurement-error matrix for Afb, these uncertainties arecombined in quadrature with the measurement variance λn.Each template provides a scan point for the χ2 function:

ðsin2 θW; χ2ðsin2 θWÞÞ. The scan points are fit to a parabolicχ2 functional form:

χ2ðsin2θWÞ ¼ χ2 þ ðsin2θW − sin2θWÞ2=σ2;

where χ2, sin2θW , and σ are parameters. The sin2θWparameter is the best-fit value of sin2 θW and σ is thecorresponding measurement uncertainty. The χ2 value,relative to 16 mass bins, is the χ2 goodness of fit.The χ2 distribution of the scan over templates from the

RESBOS NLO calculation is shown in Fig. 16. The EBA-based RESBOS calculations of Afb gives the central value of

sin2 θW . The results of the template scans are summarizedin Table III. Included in the table for comparison are twoother measurements: the CDF 2.1 fb−1 ee-pair A4 result[5], and standard model Z-pole fits from LEP-1 andSLD [7].

IX. SYSTEMATIC UNCERTAINTIES

As the forward-backward asymmetry Afb is a ratio ofcross sections, systematic uncertainties cancel out or theireffects are attenuated. The measurement of Afb employs theevent-weighting method where the simulation is used fordetector resolution unfolding and the event-weighting biascorrection. The level of the event-weighting bias correctionis kept small by limiting the measurement of Afb to akinematic region where the detector acceptance is good(jyj < 1), and the bias correction is less than 10% of thevalue of Afb. The tuning of the simulation is data driven.The small residual differences from the acceptance andmeasurement efficiencies for the simulation relative to thedata are canceled out by the event-weighting method.The systematic uncertainties contain contributions from

both the measurement of Afb and the template predictions ofAfb for various input values of sin2 θW . Both the exper-imental and prediction-related systematic uncertainties aresmall compared to the experimental statistical uncertainty.The Afb templates from the EBA-based POWHEG-BOXcalculations are used to estimate systematic uncertaintieson the sin2 θW parameter from various sources.

A. Measurement

The sources investigated are muon-charge misidentifi-cation, the momentum scale, and the background estimates.Charge misidentification is found to be negligible(Sec. V C). The total measurement systematic uncertaintyfrom the momentum scale and background isΔ sin2 θW ¼ 0.00011. The uncertainty from the back-grounds is the largest systematic uncertainty.

TABLE II. The fully corrected Afb measurement. The meas-urement uncertainties are uncorrelated bin-by-bin unfoldingestimates.

Mass bin (GeV=c2) Afb

50–80 −0.294� 0.01180–82 −0.242� 0.03382–84 −0.194� 0.02784–86 −0.135� 0.02186–88 −0.067� 0.01588–89 −0.021� 0.01489–90 0.0093� 0.008090–91 0.0427� 0.004391–92 0.0671� 0.003792–93 0.0951� 0.006293–94 0.118� 0.01194–96 0.162� 0.01396–98 0.206� 0.01498–100 0.246� 0.023100–102 0.285� 0.027>102 0.454� 0.011

Wθ2sin0.22 0.221 0.222 0.223 0.224 0.225 0.226 0.227

2 χ

20

25

30

35

40

FIG. 16 (color online). Comparison of the Afb measurementwith the RESBOS NLO templates. The triangles are the scanpoints, and the solid curve is the fit of those points to a generic χ2

functional form.


072005-16

The reconstruction-level momentum scale of both thedata and simulation are calibrated with the same techniqueto the underlying-physics scale. Thus, the reconstruction-level and physics-level mass bins used by the resolutionunfolding and the event-weighting bias correction arealigned. However, the effect from a relative offset betweenthe scales of the data and simulation is investigated. Theglobal muon-momentum scale of the data is varied todetermine the relative shifts allowed by the Z-pole massconstraint in the muon-pair invariant mass distributions ofthe experimental and simulated data. The scale shift iswell constrained by the precision of the data in the66–116 GeV=c2 mass range (Fig. 6). The resultingsystematic uncertainty from the momentum scaleis Δ sin2 θW ¼ �0.00005.Overall, the fraction of backgrounds from EWK sources

is 0.53%. In the low muon-pair invariant mass region, thelevel is approximately 5%, and the simulated event yield inthis region is slightly less than the yield of background-subtracted data. An increase in the EWK backgroundnormalization of 60% can accommodate this small differ-ence. This normalization shift is taken as the systematicuncertainty from the background normalization, and ityields Δ sin2 θW ¼ �0.00010.

B. Predictions

The QCD mass-factorization and renormalization scalesand uncertainties in the CT10 PDFs affect the Afb tem-plates. As the RESBOS calculation is chosen for the defaultAfb templates, the associated uncertainty is also included inthe overall systematic uncertainty. For the evaluation of thesystematic uncertainties, the simulation equivalent of theAfb measurement is used in template scans.Instead of calculating the series of Afb templates with

different input values of sin2 θW for each change of a QCDparameter, a simpler method is used. The sin2 θW parameteris fixed to 0.2233 for all changes of QCD parameters. Thepredicted Afb value for the mass bin m with default QCDparameters is denoted by Afbðm; 0Þ, and when the QCDparameter i is shifted, it is denoted by Afbðm; iÞ. Eachsin2 θW scan point template is offset with the difference

AfbðmÞ → AfbðmÞ þ ½Afbðm; iÞ − Afbðm; 0Þ�:

The modified templates are then used in template scans forthe best-fit value of sin2 θW . As there are no correlations ofAfb values among the mass bins, the simple bin-by-bin χ2

statistical measure is used for comparisons with thetemplates.In all QCD calculations, the mass-factorization and

renormalization scales are set to the muon-pair invariantmass. To evaluate the effects of different scales, the runningscales are varied independently by a factor ranging from 0.5to 2 in the calculations. The largest observed deviation ofthe best-fit value of sin2 θW from the default value isconsidered to be the QCD-scale uncertainty. This uncer-tainty is Δ sin2 θWðQCD scaleÞ ¼ �0.00003.The CT10 PDFs are derived from a global analysis of

experimental data that utilizes 26 fit parameters and theassociated error matrix. In addition to the global best-fitPDFs, PDFs representing the uncertainty along the eigen-vectors of the error matrix are also derived. For eacheigenvector i, a pair of PDFs are derived using 90% C.L.excursions from the best-fit parameters along its positiveand negative directions. The difference between the best-fitsin2 θW values obtained from the positive (negative) direc-tion excursion PDF and the global best-fit PDF is denoted

as δþð−Þi . The 90% C.L. uncertainty for sin2 θW is given by

the expression 12

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPiðjδþi j þ jδ−i jÞ2

p, where the sum i runs

over the 26 eigenvectors. This value is scaled down by afactor of 1.645 for the 68.3% C.L. (one standard-deviation)uncertainty yielding Δ sin2 θWðPDFÞ ¼ �0.00036.The RESBOS Afb templates are the default templates for

the extraction of sin2 θlepteff . The scan with the POWHEG-BOXor the tree templates yields slightly different values forsin2 θW . The difference, denoted as the EBA uncertainty, isΔ sin2 θWðEBAÞ ¼ �0.00012. Although the RESBOS andPOWHEG-BOX predictions are fixed-order NLO QCD cal-culations at large boson PT, they are all-orders resumma-tion calculations in the low-to-moderate PT region, whichprovides most of the total cross section. The EBA uncer-tainty is a combination of differences between the resum-mation calculations and the derived value of sin2 θW withand without QCD radiation.In summary, the total systematic uncertainties on sin2 θW

from the QCD mass-factorization and renormalization

TABLE III. Extracted values of sin2 θlepteff and sin2 θW for the EBA-based QCD templates. The PYTHIA entry is the value from the scanover non-EBA templates calculated by PYTHIA 6.4 with CTEQ5L PDFs. The uncertainties of the template scans are the measurementuncertainties (σ). Other measurements are listed in parentheses.

Template (measurement) sin2 θlepteff sin2 θW χ2

RESBOS NLO 0.2315� 0.0009 0.2233� 0.0008 21.1POWHEG-BOX NLO 0.2314� 0.0009 0.2231� 0.0008 21.4Tree LO 0.2316� 0.0008 0.2234� 0.0008 24.2PYTHIA 0.2311� 0.0008 � � � 20.8(CDF A4) 0.2328� 0.0010 0.2246� 0.0009 � � �(LEP-1þ SLD) 0.23153� 0.00016 � � � � � �


072005-17

scales, and from the CT10 PDFs is �0.00036. Allcomponent uncertainties are combined in quadrature.With the inclusion of the EBA uncertainty, the totalprediction uncertainty is �0.00038.

X. RESULTS

The values for sin2 θlepteff and sin2 θW (MW) extracted fromthe measurement of Afb using μþμ− pairs from a samplecorresponding to 9.2 fb−1 are

sin2θlepteff ¼ 0.2315� 0.0009� 0.0004

sin2θW ¼ 0.2233� 0.0008� 0.0004

MWðindirectÞ ¼ 80.365� 0.043� 0.019 GeV=c2;

where the first contribution to the uncertainties is statisticaland the second is systematic.All systematic uncertainties arecombined in quadrature, and the sources and values of theseuncertainties are summarized in Table IV. The inferred resulton sin2 θW or MW is dependent on the standard modelcontext specified in the Appendix. The sin2 θlepteff result isindependent because of its direct relationship with Afb.The measurement of sin2 θlepteff is compared with previous

measurements from the Tevatron, LHC, LEP-1, and SLD inFig. 17. The Tevatron measurements are the D0 Afb

measurement based on 5 fb−1 of integrated luminosity[4] and the CDF measurement derived from the A4

angular-distribution coefficient of ee-pairs from a samplecorresponding to 2.1 fb−1 of collisions [5]. The LHCmeasurement is the CMS analysis of Drell-Yan muon pairsfrom a sample corresponding to 1.1 fb−1 of integratedluminosity [6]. The LEP-1 and SLD measurements arefrom measurements at the Z pole. The Z-pole value is thecombination of these six measurements:

A0;lFB → 0.23099� 0.00053;

AlðPτÞ → 0.23159� 0.00041;

AlðSLDÞ → 0.23098� 0.00026;

A0;bFB → 0.23221� 0.00029;

A0;cFB → 0.23220� 0.00081;

QhadFB → 0.2324� 0.0012;

and the light-quark value is a combination of asymmetriesfrom the u, d, and s quarks [7]. The Qhad

FB measurement isbased on the hadronic charge asymmetry of all hadronicevents.TheW-boson mass inference is compared in Fig. 18 with

previous direct and indirect measurements from the

TABLE IV. Summary of the systematic uncertainties on theextraction of the weak-mixing parameters sin2 θlepteff and sin2 θW .

Source sin2 θlepteff sin2 θW

Momentum scale �0.00005 �0.00005Backgrounds �0.00010 �0.00010QCD scales �0.00003 �0.00003CT10 PDFs �0.00037 �0.00036EBA �0.00012 �0.00012

lepteffθ2sin

0.226 0.228 0.23 0.232 0.234

-1 9 fbµµCDF0.0010±0.2315

-1 2 fbeeCDF0.0010±0.2328

-1 1 fbµµCMS

0.0032±0.2287

-1 5 fbeeD00.0010±0.2309

LEP-1 and SLD: light quarks

0.0021±0.2320

LEP-1 and SLD: Z pole0.00016±0.23153

FIG. 17 (color online). Comparison of experimental measure-ments of sin2 θlepteff . “Z pole” represents the LEP-1 and SLDstandard model analysis of Z-pole measurements and “lightquarks” represents the LEP-1 and SLD results from the light-quark asymmetries; “D0 ee 5 fb−1” represents the D0 AfbðMÞanalysis; “CMS μμ 1 fb−1” represents the CMS analysis; “CDFee 2 fb−1” represents the A4 analysis; and “CDF μμ 9 fb−1”represents this analysis. The horizontal bars represent totaluncertainties.

)2W-boson mass (GeV/c80 80.1 80.2 80.3 80.4 80.5 80.6

-1 9 fbµµCDF 0.047±80.365

-1 2 fbeeCDF 0.048±80.297

NuTeV0.085±80.135

)tLEP-1 and SLD (m 0.020±80.365

TeV and LEP-2 0.015±80.385

FIG. 18 (color online). Comparison of experimental determi-nations of the W-boson mass. “TeVand LEP-2” represents directmeasurements of the W-boson mass; “LEP-1 and SLD (mt)”represents the standard model analysis of Z-pole measurements;“NuTeV” represents the indirect measurement derived fromneutrino scattering at Fermilab; “CDF ee 2 fb−1” representsthe A4 analysis; and “CDF μμ 9 fb−1” represents this analysis.The horizontal bars represent total uncertainties.


072005-18

Tevatron, NuTeV, LEP-1, SLD, and LEP-2. The indirectmeasurement from the Tevatron collider is based on the A4

angular coefficient analysis [5]. The indirect measurementfrom LEP-1 and SLD is from electroweak standard modelfits to Z-pole measurements in combination with theTevatron top-quark mass measurement [44]. The NuTeVvalue, an indirect measurement, is based on the on-shellsin2 θW parameter extracted from the measurement of theratios of the neutral-to-charged current ν and ν crosssections at Fermilab [8]. The direct measurements are fromthe Tevatron and LEP-2 [45]. The total uncertaintiesinclude both statistical and systematic uncertainties, whichare combined in quadrature. Both CDF analyses areindirect measurements of MW , and they both use the samestandard model context.

XI. SUMMARY

The angular distribution of Drell-Yan lepton pairsprovides information on the electroweak-mixing parametersin2 θW . The muon forward-backward asymmetry in thepolar-angle distribution cosϑ is governed by the A4 cosϑterm, whose A4 coefficient is directly related to the sin2 θ

lepteff

mixing parameter at the lepton vertex, and indirectly tosin2 θW . The effective-leptonic parameter sin2 θlepteff isderived from the measurement of the forward-backwardasymmetry AfbðMÞ based on the entire CDF Run II sampleof muon pairs, which corresponds to 9.2 fb−1 of integratedluminosity from pp collisions at a center-of-momentumenergy of 1.96 TeV. Calculations of AfbðMÞ with differentvalues of the electroweak-mixing parameter are comparedwith the measurement to determine the value of theparameter that best describes the data. The calculationsinclude both quantum chromodynamic and electroweakradiative corrections. The best-fit values from the compar-isons are summarized as follows:

sin2θlepteff ¼ 0.2315� 0.0010

sin2θW ¼ 0.2233� 0.0009

MWðindirectÞ ¼ 80.365� 0.047 GeV=c2:

Each uncertainty includes statistical and systematic con-tributions. Both results are consistent with LEP-1 and SLDmeasurements at the Z-boson pole. The value of sin2 θlepteff isalso consistent with the previous results from theTevatron [4,5].

ACKNOWLEDGMENTS

We thank the Fermilab staff and the technical staffs of theparticipating institutions for their vital contributions. Thiswork was supported by the U.S. Department of Energy andNational Science Foundation; the Italian Istituto Nazionaledi Fisica Nucleare; the Ministry of 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; theSwiss National Science Foundation; the A. P. SloanFoundation; the Bundesministerium für Bildung undForschung, Germany; the Korean World Class UniversityProgram, the National Research Foundation of Korea; theScience and Technology Facilities Council and the RoyalSociety, United Kingdom; the Russian Foundation forBasic Research; the Ministerio de Ciencia e Innovación,and Programa Consolider-Ingenio 2010, Spain; the SlovakR&D Agency; the Academy of Finland; the AustralianResearch Council (ARC); and the EU community MarieCurie Fellowship Contract No. 302103.

APPENDIX: ZFITTER

The input parameters to the ZFITTER radiative-correctioncalculation are particle masses, the electromagnetic fine-structure constant αem, the Fermi constant GF, the strong-interaction coupling at the Z mass αsðM2

ZÞ, and thecontribution of the light quarks to the “running” αem atthe Z mass Δαð5ÞemðM2

ZÞ (DALH5). The scale-dependentcouplings are αsðM2

ZÞ ¼ 0.118 and Δαð5ÞemðM2ZÞ ¼ 0.0275

[46]. The mass parameters areMZ ¼ 91.1875 GeV=c2 [7],mt ¼ 173.2 GeV=c2 (top quark) [44], and mH ¼125 GeV=c2 (Higgs boson). Form factors and the Z-bosontotal decay-width ΓZ are calculated.The renormalization scheme used by ZFITTER is the on-

shell scheme [13], where particle masses are on shell and

sin2θW ¼ 1 −M2W=M

2Z (A1)

holds to all orders of perturbation theory by definition. Ifboth GF and mH are specified, sin θW is not independent,and it is derived from standard model constraints that useradiative corrections. To vary the sin θW (MW) parameter,the value of GF is changed by a small amount prior tothe calculation so that the derived MW range is80.0–80.5 GeV=c2. The set of resulting MW values corre-spond to a family of physics models with standard-modellike couplings where sin2 θW and the coupling ðGFÞ aredefined by the MW parameter. The Higgs-boson massconstraint mH ¼ 125 GeV=c2 keeps the form factorswithin the vicinity of standard model fit values fromLEP-1 and SLD [7].The primary purpose of ZFITTER is to provide tables of

form factors for each model. As the form factors arecalculated in the massless-fermion approximation, theyonly 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 scattering-

amplitude ansatz is


072005-19

Aq ¼i4

ffiffiffi2

pGFM2

Z

s − ðM2Z − isΓZ=MZÞ

4Te3T

q3ρeq

× ½hejγμð1þ γ5Þjeihqjγμð1þ γ5Þjqi− 4jQejκesin2θWhejγμjeihqjγμð1þ γ5Þjqi− 4jQqjκqsin2θWhejγμð1þ γ5Þjeihqjγμjqiþ 16jQeQqjκeqsin4θWhejγμjeihqjγμjqi�;

where q ¼ u or d, the ρeq, κe, κq, and κeq are complex-valued form factors, the bilinear γ matrix terms arecovariantly contracted, and 1

2ð1þ γ5Þ is the left-handed

helicity projector in the ZFITTER convention. The κe form

factors of the Au and Ad amplitudes are not equivalent;however, at s ¼ M2

Z, they are numerically equal.The ρeq, κe, and κq form factors are incorporated into

QCD calculations as corrections to the Born-level gfA andgfV couplings:

gfV →ffiffiffiffiffiffiffiρeq

p ðTf3 − 2Qfκfsin2θWÞ and gfA →

ffiffiffiffiffiffiffiρeq

pTf3 ;

where f ¼ e or q. The resulting current-current amplitudeis similar to Aq, but the sin4 θW term contains κeκq. Thedifference is removed with the addition of this amplitudecorrection: the sin4 θW term of Aq with κeq → κeq − κeκq.Implementation details are provided in Ref. [5].

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