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.3642v1

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27Aug2007

First Run I I Measurement of the W Boson Mass

T. Aaltonen,23 A. Abulencia,24 J. Adelman,13 T. Akimoto,54 M.G. Albrow,17 B. Alvarez Gonzalez,11S. Amerio,42 D. Amidei,34 A. Anastassov,51 A. Annovi,19 J. Antos,14 G. Apollinari,17 A. Apresyan,47

T. Arisawa,56 A. Artikov,15 W. Ashmanskas,17 A. Attal,3 A. Aurisano,52 F. Azfar,41 P. Azzi-Bacchetta,42

P. Azzurri,45 N. Bacchetta,42 W. Badgett,17 A. Barbaro-Galtieri,28 V.E. Barnes,47 B.A. Barnett,25

S. Baroiant,7 V. Bartsch,30 G. Bauer,32 P.-H. Beauchemin,33 F. Bedeschi,45 P. Bednar,14 S. Behari,25

G. Bellettini,45 J. Bellinger,58 A. Belloni,32 D. Benjamin,16 A. Beretvas,17 J. Beringer,28 T. Berry,29

A. Bhatti,49 M. Binkley,17 D. Bisello,42 I. Bizjak,30 R.E. Blair,2 C. Blocker,6 B. Blumenfeld,25 A. Bocci,16

A. Bodek,48 V. Boisvert,48 G. Bolla,47 A. Bolshov,32 D. Bortoletto,47 J. Boudreau,46 A. Boveia,10

B. Brau,10 L. Brigliadori,5 C. Bromberg,35 E. Brubaker,13 J. Budagov,15 H.S. Budd,48 S. Budd,24

K. Burkett,17 G. Busetto,42 P. Bussey,21 A. Buzatu,33 K. L. Byrum,2 S. Cabrerar,16 M. Campanelli,35

M. Campbell,34 F. Canelli,17 A. Canepa,44 D. Carlsmith,58 R. Carosi,45 S. Carrillol,18 S. Carron,33

B. Casal,11 M. Casarsa,17 A. Castro,5 P. Catastini,45 D. Cauz,53 M. Cavalli-Sforza,3 A. Cerri,28

L. Cerritop

,30

S.H. Chang,27

Y.C. Chen,1

M. Chertok,7

G. Chiarelli,45

G. Chlachidze,17

F. Chlebana,17

K. Cho,27 D. Chokheli,15 J.P. Chou,22 G. Choudalakis,32 S.H. Chuang,51 K. Chung,12 W.H. Chung,58

Y.S. Chung,48 C.I. Ciobanu,24 M.A. Ciocci,45 A. Clark,20 D. Clark,6 G. Compostella,42 M.E. Convery,17

J. Conway,7 B. Cooper,30 K. Copic,34 M. Cordelli,19 G. Cortiana,42 F. Crescioli,45 C. Cuenca Almenarr,7

J. Cuevaso,11 R. Culbertson,17 J.C. Cully,34 D. Dagenhart,17 M. Datta,17 T. Davies,21 P. de Barbaro,48

S. De Cecco,50 A. Deisher,28 G. De Lentdeckerd,48 G. De Lorenzo,3 M. DellOrso,45 L. Demortier,49

J. Deng,16 M. Deninno,5 D. De Pedis,50 P.F. Derwent,17 G.P. Di Giovanni,43 C. Dionisi,50 B. Di Ruzza,53

J.R. Dittmann,4 M. DOnofrio,3 S. Donati,45 P. Dong,8 J. Donini,42 T. Dorigo,42 S. Dube,51 J. Efron,38

R. Erbacher,7 D. Errede,24 S. Errede,24 R. Eusebi,17 H.C. Fang,28 S. Farrington,29 W.T. Fedorko,13

R.G. Feild,59 M. Feindt,26 J.P. Fernandez,31 C. Ferrazza,45 R. Field,18 G. Flanagan,47 R. Forrest,7

S. Forrester,7 M. Franklin,22 J.C. Freeman,28 I. Furic,18 M. Gallinaro,49 J. Galyardt,12 F. Garberson,10

J.E. Garcia,45 A.F. Garfinkel,47 H. Gerberich,24 D. Gerdes,34 S. Giagu,50 P. Giannetti,45 K. Gibson,46

J.L. Gimmell,

48

C. Ginsburg,

17

N. Giokaris

a

,

15

M. Giordani,

53

P. Giromini,

19

M. Giunta,

45

V. Glagolev,

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D. Glenzinski,17 M. Gold,36 N. Goldschmidt,18 J. Goldsteinc,41 A. Golossanov,17 G. Gomez,11

G. Gomez-Ceballos,32 M. Goncharov,52 O. Gonzalez,31 I. Gorelov,36 A.T. Goshaw,16 K. Goulianos,49

A. Gresele,42 S. Grinstein,22 C. Grosso-Pilcher,13 R.C. Group,17 U. Grundler,24 J. Guimaraes da Costa,22

Z. Gunay-Unalan,35 C. Haber,28 K. Hahn,32 S.R. Hahn,17 E. Halkiadakis,51 A. Hamilton,20 B.-Y. Han,48

J.Y. Han,48 R. Handler,58 F. Happacher,19 K. Hara,54 D. Hare,51 M. Hare,55 S. Harper,41 R.F. Harr,57

R.M. Harris,17 M. Hartz,46 K. Hatakeyama,49 J. Hauser,8 C. Hays,41 M. Heck,26 A. Heijboer,44

B. Heinemann,28 J. Heinrich,44 C. Henderson,32 M. Herndon,58 J. Heuser,26 S. Hewamanage,4 D. Hidas,16

C.S. Hillc,10 D. Hirschbuehl,26 A. Hocker,17 S. Hou,1 M. Houlden,29 S.-C. Hsu,9 B.T. Huffman,41

R.E. Hughes,38 U. Husemann,59 J. Huston,35 J. Incandela,10 G. Introzzi,45 M. Iori,50 A. Ivanov,7

B. Iyutin,32 E. James,17 B. Jayatilaka,16 D. Jeans,50 E.J. Jeon,27 S. Jindariani,18 W. Johnson,7 M. Jones,47

K.K. Joo,27 S.Y. Jun,12 J.E. Jung,27 T.R. Junk,24 T. Kamon,52 D. Kar,18 P.E. Karchin,57 Y. Kato,40

R. Kephart,17 U. Kerzel,26 V. Khotilovich,52 B. Kilminster,38 D.H. Kim,27 H.S. Kim,27 J.E. Kim,27

M.J. Kim,17 S.B. Kim,27 S.H. Kim,54 Y.K. Kim,13 N. Kimura,54 L. Kirsch,6 S. Klimenko,18 M. Klute,32

B. Knuteson,32 B.R. Ko,16 S.A. Koay,10 K. Kondo,56 D.J. Kong,27 J. Konigsberg,18 A. Korytov,18

A.V. Kotwal,16 J. Kraus,24 M. Kreps,26 J. Kroll,44 N. Krumnack,4 M. Kruse,16 V. Krutelyov,10 T. Kubo,54

S. E. Kuhlmann,2 T. Kuhr,26 N.P. Kulkarni,57 Y. Kusakabe,56 S. Kwang,13 A.T. Laasanen,47 S. Lai,33

S. Lami,45 S. Lammel,17 M. Lancaster,30 R.L. Lander,7 K. Lannon,38 A. Lath,51 G. Latino,45 I. Lazzizzera,42

T. LeCompte,2 J. Lee,48 J. Lee,27 Y.J. Lee,27 S.W. Leeq,52 R. Lefevre,20 N. Leonardo,32 S. Leone,45

S. Levy,13 J.D. Lewis,17 C. Lin,59 C.S. Lin,17 M. Lindgren,17 E. Lipeles,9 T.M. Liss,24 A. Lister,7

D.O. Litvintsev,17 T. Liu,17 N.S. Lockyer,44 A. Loginov,59 M. Loreti,42 L. Lovas,14 R.-S. Lu,1 D. Lucchesi,42

J. Lueck,26 C. Luci,50 P. Lujan,28 P. Lukens,17 G. Lungu,18 L. Lyons,41 J. Lys,28 R. Lysak,14 E. Lytken,47

P. Mack,26 D. MacQueen,33 R. Madrak,17 K. Maeshima,17 K. Makhoul,32 T. Maki,23 P. Maksimovic,25

S. Malde,41 S. Malik,30 G. Manca,29 A. Manousakisa,15 F. Margaroli,47 C. Marino,26 C.P. Marino,24

http://arxiv.org/abs/0708.3642v1http://arxiv.org/abs/0708.3642v1http://arxiv.org/abs/0708.3642v1http://arxiv.org/abs/0708.3642v1http://arxiv.org/abs/0708.3642v1http://arxiv.org/abs/0708.3642v1http://arxiv.org/abs/0708.3642v1http://arxiv.org/abs/0708.3642v1http://arxiv.org/abs/0708.3642v1http://arxiv.org/abs/0708.3642v1http://arxiv.org/abs/0708.3642v1http://arxiv.org/abs/0708.3642v1http://arxiv.org/abs/0708.3642v1http://arxiv.org/abs/0708.3642v1http://arxiv.org/abs/0708.3642v1http://arxiv.org/abs/0708.3642v1http://arxiv.org/abs/0708.3642v1http://arxiv.org/abs/0708.3642v1http://arxiv.org/abs/0708.3642v1http://arxiv.org/abs/0708.3642v1http://arxiv.org/abs/0708.3642v1http://arxiv.org/abs/0708.3642v1http://arxiv.org/abs/0708.3642v1http://arxiv.org/abs/0708.3642v1http://arxiv.org/abs/0708.3642v1http://arxiv.org/abs/0708.3642v1http://arxiv.org/abs/0708.3642v1http://arxiv.org/abs/0708.3642v1http://arxiv.org/abs/0708.3642v1http://arxiv.org/abs/0708.3642v1http://arxiv.org/abs/0708.3642v1http://arxiv.org/abs/0708.3642v1http://arxiv.org/abs/0708.3642v1http://arxiv.org/abs/0708.3642v1http://arxiv.org/abs/0708.3642v1

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A. Martin,59 M. Martin,25 V. Martinj ,21 M. Martnez,3 R. Martnez-Ballarn,31 T. Maruyama,54

P. Mastrandrea,50 T. Masubuchi,54 M.E. Mattson,57 P. Mazzanti,5 K.S. McFarland,48 P. McIntyre,52

R. McNultyi,29 A. Mehta,29 P. Mehtala,23 S. Menzemerk,11 A. Menzione,45 P. Merkel,47 C. Mesropian,49

A. Messina,35 T. Miao,17 N. Miladinovic,6 J. Miles,32 R. Miller,35 C. Mills,22 M. Milnik,26 A. Mitra,1

G. Mitselmakher,18 H. Miyake,54 S. Moed,20 N. Moggi,5 C.S. Moon,27 R. Moore,17 M. Morello,45

P. Movilla Fernandez,28 J. Mulmenstadt,28 A. Mukherjee,17 Th. Muller,26 R. Mumford,25 P. Murat,17

M. Mussini,5 J. Nachtman,17 Y. Nagai,54 A. Nagano,54 J. Naganoma,56 K. Nakamura,54 I. Nakano,39

A. Napier,55 V. Necula,16 C. Neu,44 M.S. Neubauer,24 J. Nielsenf,28 L. Nodulman,2 M. Norman,9

O. Norniella,24 E. Nurse,30 S.H. Oh,16 Y.D. Oh,27 I. Oksuzian,18 T. Okusawa,40 R. Oldeman,29 R. Orava,23

K. Osterberg,23 S. Pagan Griso,42 C. Pagliarone,45 E. Palencia,17 V. Papadimitriou,17 A. Papaikonomou,26

A.A. Paramonov,13 B. Parks,38 S. Pashapour,33 J. Patrick,17 G. Pauletta,53 M. Paulini,12 C. Paus,32

D.E. Pellett,7 A. Penzo,53 T.J. Phillips,16 G. Piacentino,45 J. Piedra,43 L. Pinera,18 K. Pitts,24 C. Plager,8

L. Pondrom,58 X. Portell,3 O. Poukhov,15 N. Pounder,41 F. Prakoshyn,15 A. Pronko,17 J. Proudfoot,2

F. Ptohosh,17 G. Punzi,45 J. Pursley,58 J. Rademackerc,41 A. Rahaman,46 V. Ramakrishnan,58 N. Ranjan,47

I. Redondo,31 B. Reisert,17 V. Rekovic,36 P. Renton,41 M. Rescigno,50 S. Richter,26 F. Rimondi,5L. Ristori,45 A. Robson,21 T. Rodrigo,11 E. Rogers,24 S. Rolli,55 R. Roser,17 M. Rossi,53 R. Rossin,10

P. Roy,33 A. Ruiz,11 J. Russ,12 V. Rusu,17 H. Saarikko,23 A. Safonov,52 W.K. Sakumoto,48 G. Salamanna,50

O. Salto,3 L. Santi,53 S. Sarkar,50 L. Sartori,45 K. Sato,17 P. Savard,33 A. Savoy-Navarro,43 T. Scheidle,26

P. Schlabach,17 E.E. Schmidt,17 M.P. Schmidt,59 M. Schmitt,37 T. Schwarz,7 L. Scodellaro,11 A.L. Scott,10

A. Scribano,45 F. Scuri,45 A. Sedov,47 S. Seidel,36 Y. Seiya,40 A. Semenov,15 L. Sexton-Kennedy,17

A. Sfyrla,20 S.Z. Shalhout,57 M.D. Shapiro,28 T. Shears,29 P.F. Shepard,46 D. Sherman,22 M. Shimojiman,54

M. Shochet,13 Y. Shon,58 I. Shreyber,20 A. Sidoti,45 P. Sinervo,33 A. Sisakyan,15 A.J. Slaughter,17

J. Slaunwhite,38 K. Sliwa,55 J.R. Smith,7 F.D. Snider,17 R. Snihur,33 M. Soderberg,34 A. Soha,7

S. Somalwar,51 V. Sorin,35 J. Spalding,17 F. Spinella,45 T. Spreitzer,33 P. Squillacioti,45 M. Stanitzki,59

R. St. Denis,21 B. Stelzer,8 O. Stelzer-Chilton,41 D. Stentz,37 J. Strologas,36 D. Stuart,10 J.S. Suh,27

A. Sukhanov,18 H. Sun,55 I. Suslov,15 T. Suzuki,54 A. Taffarde,24 R. Takashima,39 Y. Takeuchi,54

R. Tanaka,39

M. Tecchio,34

P.K. Teng,1

K. Terashi,49

J. Thomg

,17

A.S. Thompson,21

G.A. Thompson,24

E. Thomson,44 P. Tipton,59 V. Tiwari,12 S. Tkaczyk,17 D. Toback,52 S. Tokar,14 K. Tollefson,35

T. Tomura,54 D. Tonelli,17 S. Torre,19 D. Torretta,17 S. Tourneur,43 W. Trischuk,33 Y. Tu,44 N. Turini,45

F. Ukegawa,54 S. Uozumi,54 S. Vallecorsa,20 N. van Remortel,23 A. Varganov,34 E. Vataga,36 F. Vazquezl,18

G. Velev,17 C. Vellidisa,45 V. Veszpremi,47 M. Vidal,31 R. Vidal,17 I. Vila,11 R. Vilar,11 T. Vine,30

M. Vogel,36 I. Volobouevq,28 G. Volpi,45 F. Wurthwein,9 P. Wagner,44 R.G. Wagner,2 R.L. Wagner,17

J. Wagner,26 W. Wagner,26 R. Wallny,8 S.M. Wang,1 A. Warburton,33 D. Waters,30 M. Weinberger,52

W.C. Wester III,17 B. Whitehouse,55 D. Whitesone,44 A.B. Wicklund,2 E. Wicklund,17 G. Williams,33

H.H. Williams,44 P. Wilson,17 B.L. Winer,38 P. Wittichg,17 S. Wolbers,17 C. Wolfe,13 T. Wright,34 X. Wu,20

S.M. Wynne,29 A. Yagil,9 K. Yamamoto,40 J. Yamaoka,51 T. Yamashita,39 C. Yang,59 U.K. Yangm,13

Y.C. Yang,27 W.M. Yao,28 G.P. Yeh,17 J. Yoh,17 K. Yorita,13 T. Yoshida,40 G.B. Yu,48 I. Yu,27

S.S. Yu,17 J.C. Yun,17 L. Zanello,50 A. Zanetti,53 I. Zaw,22 X. Zhang,24 Y. Zhengb,8 and S. Zucchelli5

(CDF Collaboration

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

3Institut de Fisica dAltes Energies, Universitat Autonoma de Barcelona, E-08193, Bellaterra (Barcelona), Spain4Baylor University, Waco, Texas 76798

5Istituto Nazionale di Fisica Nucleare, University of Bologna, I-40127 Bologna, Italy6Brandeis University, Waltham, Massachusetts 02254

7University of California, Davis, Davis, California 956168University of California, Los Angeles, Los Angeles, California 90024

9University of California, San Diego, La Jolla, California 9209310University of California, Santa Barbara, Santa Barbara, California 93106

11Instituto de Fisica de Cantabria, CSIC-University of Cantabria, 39005 Santander, Spain12Carnegie Mellon University, Pittsburgh, PA 15213

13Enrico Fermi Institute, University of Chicago, Chicago, Illinois 60637

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14Comenius University, 842 48 Bratislava, Slovakia; Institute of Experimental Physics, 040 01 Kosice, Slovakia15Joint Institute for Nuclear Research, RU-141980 Dubna, Russia

16

Duke University, Durham, North Carolina 2770817Fermi National Accelerator Laboratory, Batavia, Illinois 6051018University of Florida, Gainesville, Florida 32611

19Laboratori Nazionali di Frascati, Istituto Nazionale di Fisica Nucleare, I-00044 Frascati, Italy20University of Geneva, CH-1211 Geneva 4, Switzerland

21Glasgow University, Glasgow G12 8QQ, United Kingdom22Harvard University, Cambridge, Massachusetts 02138

23Division of High Energy Physics, Department of Physics,University of Helsinki and Helsinki Institute of Physics, FIN-00014, Helsinki, Finland

24University of Illinois, Urbana, Illinois 6180125The Johns Hopkins University, Baltimore, Maryland 21218

26Institut fur Experimentelle Kernphysik, Universitat Karlsruhe, 76128 Karlsruhe, Germany27Center for High Energy Physics: Kyungpook National University,

Taegu 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

28Ernest Orlando Lawrence Berkeley National Laboratory, Berkeley, California 9472029University of Liverpool, Liverpool L69 7ZE, United Kingdom

30University College London, London WC1E 6BT, United Kingdom31Centro de Investigaciones Energeticas Medioambientales y Tecnologicas, E-28040 Madrid, Spain

32Massachusetts Institute of Technology, Cambridge, Massachusetts 0213933Institute of Particle Physics: McGill University, Montreal,

Canada H3A 2T8; and University of Toronto, Toronto, Canada M5S 1A734University of Michigan, Ann Arbor, Michigan 48109

35Michigan State University, East Lansing, Michigan 4882436University of New Mexico, Albuquerque, New Mexico 87131

37Northwestern University, Evanston, Illinois 6020838The Ohio State University, Columbus, Ohio 4321039

Okayama University, Okayama 700-8530, Japan40Osaka City University, Osaka 588, Japan41University of Oxford, Oxford OX1 3RH, United Kingdom

42University of Padova, Istituto Nazionale di Fisica Nucleare,Sezione di Padova-Trento, I-35131 Padova, Italy

43LPNHE, Universite Pierre et Marie Curie/IN2P3-CNRS, UMR7585, Paris, F-75252 France44University of Pennsylvania, Philadelphia, Pennsylvania 19104

45Istituto Nazionale di Fisica Nucleare Pisa, Universities of Pisa,Siena and Scuola Normale Superiore, I-56127 Pisa, Italy46University of Pittsburgh, Pittsburgh, Pennsylvania 15260

47Purdue University, West Lafayette, Indiana 4790748University of Rochester, Rochester, New York 14627

49The Rockefeller University, New York, New York 1002150Istituto Nazionale di Fisica Nucleare, Sezione di Roma 1,

University of Rome La Sapienza, I-00185 Roma, Italy51Rutgers University, Piscataway, New Jersey 08855

52Texas A&M University, College Station, Texas 7784353Istituto Nazionale di Fisica Nucleare, University of Trieste/ Udine, Italy

54University of Tsukuba, Tsukuba, Ibaraki 305, Japan55Tufts University, Medford, Massachusetts 02155

56Waseda University, Tokyo 169, Japan57Wayne State University, Detroit, Michigan 48201

58University of Wisconsin, Madison, Wisconsin 5370659Yale University, New Haven, Connecticut 06520

We describe a measurement of the W boson mass mW using 200 pb1 of

s=1.96 TeV ppcollision data taken with the CDF II detector. With a sample of 63,964 W e candidates and51,128W candidates, we measure mW= [80.413 0.034(stat) 0.034(sys) = 80.413 0.048]GeV/c2. This is the single most precise mW measurement to date. When combined with other

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measured electroweak parameters, this result further constrains the properties of new unobservedparticles coupling to W and Z bosons.

PACS numbers: 13.38.Be, 14.70.Fm, 13.85.Qk, 12.15.Ji

I. INTRODUCTION

The discovery of the W and Zbosons in 1983 [1]confirmed a central prediction of the unified modelof electromagnetic and weak interactions [2]. Ini-tialW andZboson mass measurements verified thetree-level predictions of the theory, with subsequentmeasurements probing the predictedO(3 GeV/c2)[3,4] radiative corrections to the masses. The currentknowledge of these masses and other electroweak pa-rameters constrains additional radiative correctionsfrom unobserved particles such as the Higgs bosonor supersymmetric particles. These constraints arehowever limited by the precision of theWboson massmW, making improved measurements ofmW a highpriority in probing the masses and electroweak cou-plings of new hypothetical particles. We describe inthis article the single most precise mWmeasurement[5] to date.

The W boson mass can be written in terms ofother precisely measured parameters in the on-shellscheme as[4]:

m2W = 3

cEM

2GF(1 m2W/m2Z)(1 r), (1)

where EM is the electromagnetic coupling at therenormalization energy scale Q = mZc

2, GF is theFermi weak coupling extracted from the muon life-time, mZ is the Zboson mass, and r includes allradiative corrections. Fermionic loop corrections in-crease the W boson mass by terms proportional to

With visitors from aUniversity of Athens, 15784 Athens,

Greece,

b

Chinese Academy of Sciences, Beijing 100864, China,cUniversity of Bristol, Bristol BS8 1TL, United Kingdom,dUniversity Libre de Bruxelles, B-1050 Brussels, Belgium,eUniversity of California, Irvine, Irvine, CA 92697, fUniversityof California Santa Cruz, Santa Cruz, CA 95064, gCornellUniversity, Ithaca, NY 14853, hUniversity of Cyprus, NicosiaCY-1678, Cyprus, iUniversity College Dublin, Dublin 4, Ire-land, j University of Edinburgh, Edinburgh EH9 3JZ, UnitedKingdom, k University of Heidelberg, D-69120 Heidelberg, Ger-many, lUniversidad Iberoamericana, Mexico D.F., Mexico,mUniversity of Manchester, Manchester M13 9PL, England,nNagasaki Institute of Applied Science, Nagasaki, Japan,oUniversity de Oviedo, E-33007 Oviedo, Spain, p Queen MarysCollege, University of London, London, E1 4NS, England,qTexas Tech University, Lubbock, TX 79409, r IFIC(CSIC-Universitat de Valencia), 46071 Valencia, Spain,

ln(mZ/mf) for mf mZ [4], while the loop con-taining top and bottom quarks (Fig.1) increasesmWaccording to [6]:

rtb = c

3

3GFm2W8

22(m2Z m2W)

m2t + m

2b

2m2tm2b

m2t m2bln(m2t/m

2b)

,

(2)

where the second and third terms can be neglectedsince mt mb. Higgs loops (Fig. 2) decrease mWwith a contribution proportional to the logarithm ofthe Higgs mass (mH). Contributions from possiblesupersymmetric particles are dominated by squarkloops (Fig. 3) and tend to increase mW. Gener-ally, the lighter the squark masses and the larger thesquark weak doublet mass splitting, the larger thecontribution to mW. The total radiative correctionfrom supersymmetric particles can be as large as sev-eral hundred MeV/c2 [7].

Table I [8] shows the change in mW for +1

changes in the measured standard model input pa-rameters and the effect of doubling mH from 100GeV/c2 to 200 GeV/c2. In addition to the listedparameters, a variation of1.7 MeV/c2 on the pre-dictedmWarises from two-loop sensitivity to s, e.g.via gluon exchange in the quark loop in Fig.1. The-oretical corrections beyond second order, which haveyet to be calculated, are estimated to affect the mWprediction by4 MeV/c2 [8].

+W +W

t

b

FIG. 1: The one-loop contribution to the W boson massfrom top and bottom quarks.

The uncertainties on the mW prediction can becompared to the 29 MeV/c2 uncertainty on the worldaverage from direct mW measurements (Table II),

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W W

H

W W

H

FIG. 2: Higgs one-loop contributions to the W bosonmass.

W W

q~

W W

q~

q~

FIG. 3: One-loop squark contributions to the W bosonmass.

which include results from four experiments, ALEPH[12], DELPHI [15], L3 [14], and OPAL [13], study-ing

s = 161 209 GeV e+e collisions at the

Large Electron Positron collider (LEP), and from twoexperiments, CDF [16] and D [17, 18], studyings= 1.8 TeV pp collisions in Run I of the Fermilab

Tevatron. The current experimentalmW uncertaintyis a factor of two larger than the uncertainty fromradiative corrections, excluding the Higgs contribu-tion (TableI). The Higgs mass constraint extractedfrom the Wboson mass is thus limited by the directmWmeasurement. The precisemWmeasurement de-scribed in this article has a significant impact on theworld averagemW.

II. OVERVIEW

A measurement ofmW at ap pcollider [21] is com-plementary to that at an e+e collider. Individual u(d) quarks inside the proton can interact with d (u)quarks inside the anti-proton (or vice versa), allow-ing singleW+ (W) boson production, which is notpossible at an e+e collider. In addition, pp collid-ers have higher center of mass energies and Wbosonproduction cross sections. This provides high statis-tics for the leptonic decays of the W boson, whichare studied exclusively because of the overwhelminghadronic-jet background in the quark decay channels.

Parameter Shift mW Shift

(MeV/c2)

ln mH= +0.693 -41.3

mt= +1.8 GeV/c2 [9] 11.0

EM(Q= mZ c2) = +0.00035 [10] -6.2

mZ = +2.1 MeV/c2 [11] 2.6

TABLE I: The effect on mW of +1 increases of the in-put parameters dominating the uncertainty on the mWprediction. Since the Higgs boson has not been observed,we show the effect of doubling the Higgs boson mass from100 GeV/c2 to 200 GeV/c2 [8].

Experiment mW (GeV/c2)

ALEPH [12] 80.440 0.051OPAL [13] 80.416 0.053L3 [14] 80.270 0.055DELPHI [15] 80.336 0.067CDF Run I [16] 80.433 0.079D Run I [17, 18] 80.483 0.084LEP Average [19] 80.376 0.033Tevatron Run I Average [20] 80.456 0.059World Average 80.392 0.029

TABLE II: Direct measurements of the W boson mass,the preliminary combined LEP average, the combinedTevatron Run I average, and the preliminary world av-erage.

The leptonic decays of singly producedZbosons pro-vide important control samples, since both leptonsfromZboson decay are well measured. The produc-tion and decay uncertainties on the measurement ofmW fromp pand e

+e collider data are almost com-pletely independent[22].

We present in this Section an overview ofW andZ boson production at the Tevatron, a descriptionof the coordinate definitions and symbol conventionsused for this measurement, and a broad discussion ofour mWmeasurement strategy.

A. W and ZBoson Production and Decay

W and Zbosons are produced in

s = 1.96 TeVpp collisions primarily through schannel annihila-tion of valence u and/or d quarks (Fig. 4), with asmaller O(20%) contribution from sea quarks. Thequark (antiquark) has a fraction xp (xp) of the pro-

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tons (antiprotons) total momentum, producing aW

or Z boson at center of mass energy

s

Q equal

to its mass times c2. The rate of production can bepredicted from two components: (1) the momentumfraction distributions of the quarks, fq(x, Q

2), whichare determined from fits to world data [23, 24]; and(2) a perturbative calculation of the qq W or Zboson process[25].

d (u)

u

u (d)

pEpx

u

u

d

pEpx

)0

(Z+WQ

+l

)-(l

FIG. 4: Leading-order annihilation of a quark and an-tiquark inside the proton and antiproton, respectively,producing a W+ or Z0 boson. The quark (antiquark)has energy xpEp (xpEp), where Ep (Ep) represents thetotal proton (antiproton) energy. The production occursat a partonic center-of-mass energy Q. Theuu Z0 anddu W processes are similar.

W and Z bosons can decay to lepton or quarkpairs. Decays to quark pairs are not observablegiven the large direct qq background, and decays to +hadrons are not as precisely measured asboson decays to electrons or muons. For these rea-sons we restrict ourselves to the direct electronic andmuonic decays (W e, W , Z ee, andZ ), with the corresponding decays to lep-tons considered as backgrounds to these processes(Section VIII). The branching ratio for each lep-tonic decay W l (Z ll) is11% (3.3%), andthe measured cross section times branching ratio is(2749 174) pb [(254.9 16.2) pb][26].

B. Conventions

We use both Cartesian and cylindrical coordinatesystems, in which +z points in the direction of theproton beam (east) and the origin is at the centerof the detector. In the right-handed Cartesian coor-dinate system, +x points north (outward from thering) and +y points upwards; in the cylindrical sys-tem, is the azimuthal angle andr is the radius fromthe center of the detector in the x y plane. Therapidity y = 12ln[(Epzc)/(E+ pzc)] is additive

lTp

Tp

Tu

||u

u

FIG. 5: A W boson event, with the recoil hadron mo-mentum (uT) separated into axes parallel (u||) and per-pendicular (u) to the charged lepton.

under Lorentz boosts along the z axis. For masslessparticles, this quantity is equal to the pseudorapidity = ln[tan(/2)], where is the polar angle withrespect to thez axis. All angles are quoted in radiansunless otherwise indicated.

Because the interacting quarks longitudinal mo-menta pz are not known for each event, we gener-ally work with momenta transverse to the beam line.The interacting protons and antiprotons have no nettransverse momentum. Electron energy (muon mo-mentum) measured using the calorimeter (tracker) isdenoted as E (p), and the corresponding transverse

momenta pT are derived using the measured trackdirection and neglecting particle masses. The eventcalorimetricpT, excluding the lepton(s), is calculatedassuming massless particles using calorimeter towerenergies (SectionIIIA 2) and the lepton productionvertex, and provides a measurement of the recoil mo-mentum vector uT. The component of recoil pro-

jected along the lepton direction is denoted u|| andthe orthogonal component is u (Fig.5). The trans-verse momentum imbalance in a W boson event isa measure of the neutrino transverse momentum p Tand is given by p/T =(p lT +uT), where p lT is themeasured charged lepton transverse momentum.

When electromagnetic charge is not indicated,both charges are considered. We use units where= c 1 for the remainder of this paper.

C. Measurement Strategy

The measurement of the final state from W ldecays involves a measurement of p lT and the totalrecoil uT. The neutrino escapes detection and theunknown initial partonic pz precludes the use ofpzconservation in the measurement. The boson invari-ant mass is thus not reconstructable; rather, the 2-

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dimensional transverse massmTis used in themWfit:

mT =

2p lTp/T(1 cos), (3)

where is the angle in the transverse plane betweenthe leptons, whose masses are negligible. The fit tothe mT distribution provides the statistically mostprecise measurement ofmW.

The charged lepton, which can be measured pre-cisely, carries most of the observable mass informa-tion in the event. We calibrate the muon momen-tum using high statistics samples of the meson de-cays J/ and , which are fully re-constructable and have well known masses. This re-

sults in a precise track momentum calibration, whichwe transfer to the calorimeter with a fit to the ra-tio of calorimeter energy to track momentum (E/p)of electrons from W boson decays. The accuracy ofthese calibrations is demonstrated by applying themto measurements of the Zboson mass in the muonand electron decay channels. We then incorporatethe knownZboson mass as an additional calibrationconstraint.

The other directly measurable quantity needed forthe calculation ofmTis the recoil transverse momen-tumuT. Since theWandZbosons are produced at asimilarQ2, they have similar recoil distributions. Weuse the leptons from the Zboson decay to measure

thepTof the Zboson. We then calibrate our modelofuT by measuring the balance between the recoiland Z boson pT. The Z boson statistics are suffi-cient to perform a recoil calibration to 1% accuracy,which leads to a systematic uncertainty commensu-rate with other uncertainties onmW.

To accurately model the shape of the mT distri-bution, we use a fast Monte Carlo simulation of the

pp W l process including the recoil and thedetector response. The custom fast simulation allowsflexibility in parametrizing the detector response andin separating the effects of the detector model com-ponents. We use a binned likelihood to fit the mea-

sured mT distributions to templates (Section II D)generated from the fast simulation, with mW as thefree parameter. AllmWand lepton energy scale fitsare performed with this procedure.

Though less statistically precise, the plT andp/T distributions provide additional information onthe W boson mass and are used as important testsof consistency. We separately fit these distributionsformWand combine all fits in our final result.

During the measurement process, all W bosonmass fits were offset by a single unknown randomnumber chosen from a flat distribution in the range[-100,100] MeV. The fit result was thus blinded to

the authors until the analysis was complete [27]. Thefinal measured mW and its uncertainty have not

changed since the random offset was removed fromthe fit results.

We give a brief overview of the template likelihoodfitting procedure in SectionIID. SectionIIIdescribesthe detector and the fast detector simulation used inthe analysis. The W boson measurement samplesare defined in SectionIV. We describe the precisionmeasurements of muons and electrons in Sections VandVI,respectively. These sections include event se-lection, calibration, and resolution studies from thedilepton andWboson data samples. Measurement ofthe recoil response and resolution is presented in Sec-tionVII. The backgrounds to the Wboson sample

are discussed in SectionVIII. Theoretical aspects ofWandZboson production and decay, including con-straints from the current data sample, are describedin SectionIX. We present the Wboson mass fits andcross-checks in SectionX. Finally, in SectionXIweshow the result of combining our measurement withprevious measurements, and the corresponding impli-cations on the predicted standard model Higgs bosonmass.

D. Template Likelihood Fits

All the fits involving mass measurements and theenergy scale (SectionsV,VI, andX) are performedwith a template binned likelihood fitting procedure.A given distribution to be fit is generated as a discretefunction of the fit parameter, using the fast simula-tion. These simulated distributions are referred toas templates. For each value of the fit parameter,the simulated distribution is compared to the datadistribution and the logarithm of a binned likelihoodis calculated. The binned likelihood is the Poissonprobability for each bin to contain the ni observeddata events givenmiexpected events, multiplied overtheNbins in the fit range:

L = Ni=1

eminmiini!

. (4)

We calculate the logarithm of the likelihood using theapproximation ln n! (n+ 1/2)ln(n+ 1) n:

ln L Ni=1

[niln mi mi (ni+ 1/2) ln(ni+ 1 ) + ni].

(5)The best-fit value of the parameter maximizes thelikelihood (or equivalently minimizes ln L), and the1values are those that increase ln L by 1/2. The

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approximation for ln n! only affects the shape of thelikelihood about the minimum and not the position of

the minimum. The procedure is validated by fittingsimulated data (pseudoexperiments) and no bias isfound. We symmetrize the uncertainty by taking halfthe difference between the +1 and1 values. Forthe E/p fits in the Wboson sample, we reduce theeffect of finite template statistics by fitting lnL toa parabola, and extracting the best-fit value and theuncertainty from this parabola.

III. DETECTOR AND MODEL

The CDF II detector [26,28] is well suited for themW measurement. Its high-resolution tracker andcalorimeter measure individual charged lepton mo-menta fromW andZboson decays with a resolutionof 2%. It has similar acceptance and resolution forcentral electrons and muons, giving the two channelssimilar weight in a combined mass measurement.

A. Detector Components

The CDF II detector (Fig. 6) is a multi-purpose de-tector consisting of: an inner silicon tracker designed

to measure the production vertex of charged particleswith high precision; an outer tracking drift cham-ber to measure charged particle momenta; a solenoidto provide a uniform 1.4 T magnetic field insidethe trackers; electromagnetic calorimeters to containand measure electron and photon showers; hadroniccalorimeters for hadron energy measurements; and amuon system to detect muons escaping the calorime-ters. The detector information is read out on-lineand saved for later analysis when event topologiesconsistent with a particular physics process (or classof processes) are selected. The read-out decision ismade with a fast three-level trigger system that hashigh efficiency for selecting the W and Z bosons to

be used in the offline analysis.

1. Tracking System

The silicon tracker (Fig.7) consists of three sepa-rate detectors: Layer 00, SVX II, and ISL. Layer 00 isa single layer of 300m thick sensors attached to thebeam pipe at a radius of 1.3 cm. Five additional lay-ers of sensors at radii ranging from 2.5 cm to 10.6 cmcomprise SVX II. Surrounding these sensors are portcards, which transport deposited charge information

from the silicon wafers to the readout system. The in-termediate silicon layers (ISL) are located at radii of

20.2 cm and 29.1 cm. The SVX II is segmented longi-tudinally into three barrels in the region |z|

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FIG. 6: A cut-away view of a section of the CDF detector. The slice is along the y -axis at x = 0 cm.

calorimeter covers|| < 1.1 and is split at the cen-ter into two separate barrels covering + and.Each barrel consists of 24 azimuthal wedges ofsize 0.26 radians (15) with ten projective towers ofsize 0.11. To allow a pathway for the solenoidcryogenic tubes, a two-tower region is removed, cor-responding to 0.77<

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FIG. 7: End view of the silicon detector. The inner-most layer (Layer 00) is attached to the beam pipe, andis surrounded by five concentric layers of silicon wafers(SVX II). The outermost layers are the intermediate sil-icon layers (ISL), which sit just inside the outer trackingchamber.

mentation is 0.13 radians up to|| = 2.1, and thenbroadens to 0.26 radians. The two furthest forwardplug towers cover the || regions 2.63.0 and 3.03.6,while the remaining towers have a size = 0.1.

3. Muon Detectors

The muon systems relevant for the W mass mea-surement cover the region || 1. The centralmuon detector (CMU) and the central muon upgrade(CMP) cover || 0.6, while the central muon exten-sion (CMX) covers 0.6< || 1.

The CMU detector[33] is located at the outer edgeof the CHA, 347 cm from the z axis. The CMU issegmented into 15 azimuthal wedges containing fourlayers of proportional drift chambers that cover 12.6.

The maximum drift time within a chamber is 800 ns,about twice as long as the 396 ns spacing between

pp crossings. CMU information must therefore becombined with reconstructed COT particle tracks todetermine the appropriate p p crossing.

Because the total thickness of the central calorime-ter is about five interaction lengths, approximately0.5% of high-momentum pions reach the CMU. Toreduce this background, the CMP detector is locatedbehind an additional 60 cm of steel. The CMP hasa similar construction to the CMU, with the excep-tion that wider drift chambers are used to cover thesame solid angle, resulting in a maximum drift time

of 1.8s rather than 800 ns.The CMX detector [34] consists of eight drift

chamber layers beyond both the calorimeter and thesteel detector support structure (6 10 interactionlengths). The CMX regions used in this analysisare45 < < 75 and 105 < < 225. Newdetectors for Run II cover much of the remaining region, but were not fully commissioned for the data-taking period of this analysis. Scintillator detectors(CSX) at the inner and outer surfaces of the CMXprovide timing information to the trigger to separatecollision particles from other sources such as beamhalo or cosmic rays.

4. Trigger System

The trigger consists of three stages with progres-sively greater sophistication of event reconstruction.The first stage is hardware-based, the second a mixof hardware and software, and the third a farm ofprocessors performing full event reconstruction.

The first trigger stage, level 1, includes tracker,calorimeter, and muon reconstruction. The chargedparticle track reconstruction is performed with theextremely fast tracker (XFT) [35] based on the fouraxial COT superlayers. A track segment is recon-structed in a given superlayer if at least 11 of the 12

sense wires [36] in a wide road have charge deposi-tion above a given threshold (hits). The list of seg-ments from the full tracker is compared to predefinedgroups of segments expected from charged particlesabove a given momentum threshold. When matchesare found, track candidates are created and passedto the track extrapolator (XTRP) [37]. The XTRPdetermines the expected positions of the tracks inthe calorimeter and muon detectors, for the purposeof forming electron and muon candidates.

The calorimeter reconstruction at level 1 definesseparate electromagnetic and hadronic trigger tow-ers as tower pairs adjacent in . The tower pT is

calculated assuming a collision vertex z = 0 and anelectron candidate is formed if the ratio of hadronic toelectromagnetic energy (Had/EM) in a trigger toweris less than 1/8. The high-momentum electron triggerused in this analysis requires a level 1 trigger towerwith electromagneticpT >8 GeV matched to a trackwith pT > 8 GeV, and drops the Had/EM require-ment for electromagneticpT >14 GeV.

Level 1 muon reconstruction includes a pT esti-mate within the CMU and CMX chambers fromthe relative timing of the hits in different layers.The CMU track segments are combined with recon-structed CMP track segments to create CMUP

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FIG. 8: End view of a section of a central outer tracker (COT) endplate. The COT consists of eight concentricsuperlayers, separated azimuthally into cells, each containing 12 sense wires and sandwiched by field sheets. Theendplates contain precision-machined slots where each cells sense wires and field sheets are held under tension. Theradius at the center of each superlayer is shown in cm.

muon candidates. For the majority of the data CMX

candidates also require local CSX hits consistent withparticles originating from the collision. For our WandZboson samples we use a muon trigger that re-quires CMU or CMX pT > 6 GeV matched to anXFT track with pT > 4 GeV (CMUP) or pT > 8GeV (CMX).

The level 2 calorimeter reconstruction uses a moresophisticated clustering algorithm for electromag-netic objects. This improves energy measurementresolution and allows a higher threshold (pT > 16GeV) to be applied. To reduce rates, the XFTtrack requirement for CMUP candidates was raisedtopT >8 GeV for most of the data-taking period.

At level 3, approximately 300 dual processor com-puters allow full track pattern recognition, muon re-construction, and calorimeter clustering. Variablesused to select electrons at level 3 are the lateralshower profile, Lshr (SectionIV B), and the distancebetween CES zand thez-position of the track extrap-olated to the CES (z). TheLshrvariable quantifiesthe difference between the measured energies of tow-ers neighboring the electron in and the expectedenergies determined from electron test beam data.The trigger requirements ofLshr< 0.4 and|z| 18 GeV and track

pT >9 GeV. For efficiency studies we use a separatetrigger that requires electromagnetic pT > 25 GeVandp/T

L3 >25 GeV, but has no quality requirementsat level 3 and no trigger track requirements. At level3, p/T

L3 is defined as the negative of the vector sum

of the transverse momenta in all calorimeter towers.The high-momentum muon trigger requires a COTtrack with pT > 18 GeV matched to a CMUP orCMX track segment.

5. Luminosity Detector

The small-angle Cherenkov luminosity counters(CLC) [38] are used to measure the instantaneousand integrated luminosity of our data samples. TheCLC consists of two modules installed around thebeampipe at each end of the detector, providing cov-erage in the regions 3.6

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(SectionVII).

B. Detector Model

We use a parametrized model of the detector re-sponse to electrons, muons, and the hadronic recoil.The model is incorporated into a custom fast simu-lation that includes lepton and recoil reconstruction,event selection, and fit template generation. The sim-ulation provides both flexibility in determining the ef-fects of various inputs, and computing speed to allowfrequent high-statistics studies. A sample of O(107)

events can be generated using a single-processor ma-chine in one day. This is several orders of magnitudemore than the O(103) events that can be producedwith the standard geant-based CDF simulation[40][41].

We describe in this section the simulation of elec-trons and muons. Fits to the data that determine thevalues of some of the model parameters are describedin Secs. VandVI. The detector model of hadronicrecoil response and resolution is discussed in Sec. VII.

The model components common to muons andelectrons are: ionization energy loss and multiple

scattering in the beam pipe and tracker volume;parametrized track hit resolutions and efficiencies;and track reconstruction. We describe these com-ponents in the muon simulation overview, and thendiscuss the electron- and photon-specific simulation.

1. Muon Simulation

Muon and electron tracks are reconstructed usingonly COT hit and beam position information (Sec-

tionIV). Thus, the simulation of the silicon detectorconsists entirely of energy loss and multiple scatter-ing. In the COT, hit resolutions and efficiencies areadditionally simulated, and track reconstruction isperformed. The total measured muon EM calorime-ter energy is simulated by combining the minimum-ionizing energy deposition with energy from final-state photon radiation (SectionIX D) and the recoiland underlying event[39]. Finally, the detector fidu-ciality of muons is calculated using a map of the muondetector geometry as a function of and . The mapis extracted from a full geant-based simulation ofthe CDF II detector[40,41].

Ionization Energy Loss

The differential ionization energy loss of muons andelectrons in the tracking system is simulated accord-ing to the Bethe-Bloch equation[11]:

dEdx

= KZ

A2

1

2ln

2me2Tmax

(1 2)I2 2

2

, (6)

where K = 4NAr2eme, NA is Avogadros number,

re is the classical electron radius, Z(A) is the atomic(mass) number, is the particle velocity, I is themean excitation energy, Tmax is the maximum kine-matic energy that can be transferred to a free electronin a single collision, and is the material-dependent

density effect as a function of[11]. When calculat-ing the effect of, we take the material to be siliconthroughout.

To calculate muon energy loss in the material up-stream of the COT (r < 40 cm), we use a three-dimensional lookup table of the material propertiesof the beam pipe, the silicon detector, and the wallof the alumnium can at the inner radius of the COT.The lookup table determines the appropriate Z/AandIvalues, along with the radiation lengthX0(Ap-pendixA), for each of 32 radial layers. Except for theinner and outer layers, the map is finely segmentedlongitudinally and in azimuth to capture the material

variation in the silicon detector[42]. Inside the COTfiducial volume we calculate the energy loss betweeneach of the 96 radial sense wires.

The energy loss model is tuned using the data. Weapply a global correction factor of 0.94 to the calcu-lated energy loss in the material upstream of the COTin order to obtain a J/ mass measurementthat is independent of the mean inverse momentumof the decay muons (SectionV B 3).

Multiple Coulomb Scattering

Multiple Coulomb scattering in the beampipe, sil-icon detector, and COT affects the resolution of thereconstructed track parameters for low-momentumtracks. We model the scattering using a Gaussiandistribution for 98% of the scatters[43] with an an-gular resolution defined by

=13.6 MeV

p

x/X0, (7)

where x is the thickness of the layer and X0 is thelayers radiation length (SectionIII B2). Simulationof multiple scattering is implemented for each radial

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layer of the three-dimensional lookup table and be-tween each COT layer.

Based on the results of low-energy muon scatter-ing data [44], we model the non-Gaussian wide-anglescatters by increasing by a factor of 3.8 for 2% ofthe scatters.

COT Simulation and Reconstruction

The charged track measurement is modeled witha full hit-level simulation of the charge deposition inthe COT and a helical track fit. The parameter reso-lution of reconstructed tracks is affected by the indi-vidual hit resolution, and by the distribution of the

number of hits (Nhit) used in the fit[45].We tune the COT hit resolution using the width of

the mass distribution reconstructed withnon-beam-constrained tracks. The tuned value of[1503(stat)]m is consistent with the 149 m RMSof the observed hit residual distribution for the muontracks in Z data. We use a 150 m hit resolu-tion for the simulation of the , W, and Z bosons.

We use a dual-resolution model to describe the nar-rower mass peak in the high-statistics J/ sample, where the muons generally have lower mo-menta than the other samples. The J/ mass peakwidth is particularly sensitive to multiple scattering

and relative energy loss, and our hit-resolution modelcompensates for any mismodeling that affects thepeak width. We find that a single-hit resolution of155m applied to 70% of the tracks and 175 m ap-plied to the remaining 30% adequately describes thewidth and lineshape of the J/ mass peak.

To describe the Nhit distribution, we use a dual-hit-efficiency model, the larger one applied to the ma-

jority of the tracks. The lower efficiency accounts forevents with high COT occupancy, where fewer hitsare attached to reconstructed tracks. The two pa-rameters are tuned to match the mean and RMS ofthe data Nhit distributions. We independently tune

these parameters for theJ/ sample, the sample,and the W andZ boson samples.COT hit positions from a charged track are used

to reconstruct a helix with a 2-minimization proce-dure. The axial helix parameters[46] are the impactparameter with respect to the nominal beam position,d0, the azimuthal angle at the closest approach to thebeam, 0, and the curvature of the track, c, definedto be (2R)1, where R is the radius of curvature.The stereo helix parameters are the longitudinal po-sition at the closest approach to the beam, z0, andthe cotangent of the polar angle, cot .

When optimizing resolution of lepton tracks from

prompt resonance decays, we constrain the helix tooriginate from the location of the beam. The trans-

verse size of the beam is 30 m at z = 0 c mand increases to 50 60 m at|z| = 40 cm [47].For simplicity we assume an average beam size of[39 3(stat)] m, which is determined from a fitto the width of the Z mass peak. Thebeam constraint improves the intrinsic fractional mo-mentum resolution by about a factor of three, topT/pT 0.0005pT/GeV.

We perform a track fit on our simulated hits in thesame manner as the data. The hits are first fit to ahelix without a beam constraint; hits with large resid-uals (>600m) are dropped from the track (in orderto remove spurious hits added in data pattern recog-

nition); and the track is fit again with an optionalbeam constraint. This option is applied to promptlepton tracks fromWandZboson decays, but not totracks fromJ/ decays, approximately 20% of whichare not prompt. The prompt muons from decaysare fit twice, both with and without the beam con-straint, as a consistency check.

Calorimeter Response

Muons deposit ionization energy in the calorimeter.We simulate a muons EM energy deposition using

a distribution taken from cosmic ray muons passingthrough the center of the detector, in events withno other track activity. An additional contributioncomes from energy flow into the calorimeter from theunderlying event [39]. We model this energy using adistribution taken from W data events, usingtowers separated in azimuth from the muon.

Muons with a CES z position within 1.58 cm ofa tower boundary typically deposit energy in twocalorimeter towers. We use this criterion in the sim-ulation to apply the underlying event and final-statephoton radiation (SectionIX D) contributions for oneor two towers. The simulated underlying event en-

ergy includes its dependence on u|| and u (Fig. 5),and on the tower position of the muon when itcrosses the CES (SectionVIIB).

Detector Fiduciality

The CMUP and CMX muon systems do not havecomplete azimuthal or polar angle coverage. We cre-ate an map of each muon detectors coverageusing muons simulated[41] with a detector geometrybased on geant [40]. We use the map in the fastsimulation to determine the fiduciality of a muon at

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a given position.We incorporate the relative efficiency of the CMUP

to CMX triggers in the fast simulation by matchingthe ratio of CMUP to CMX events in the W data (SectionIV A).

2. Electron and Photon Simulation

The dominant calibration of the calorimeter energymeasurement Eof electrons uses their track momenta

p and a fit to the peak of the E/p distribution. Anadditional calibration results from a mass fit to theZboson resonance and reduces the calibration uncer-tainty by 20% relative to the E /p calibration alone.

The E/p method relies on an accurate modelingof radiative effects that reduce the track momentummeasured in the COT. A given electron loses 20%of its energy through bremsstrahlung radiation in thesilicon detector, and this process has the most sig-nificant impact on the E/p calibration. The totalamount of silicon detector material is tuned with datausing highly radiative electrons (Section VI A). Weadditionally model processes that affect the shapeof the E/p distribution: photon conversion in thetracker; energy loss in the solenoid and the time-of-flight system; electromagnetic calorimeter responseand resolution; and energy loss into the hadroniccalorimeter. The models of ionization energy loss andmultiple scattering in the tracker, as well as the COTtrack simulation and reconstruction, are the same asfor muons (SectionIIIB 1).

Bremsstrahlung

The differential cross section for an electron of en-ergyEe to radiate a photon of energyE is given bythe screened Bethe-Heitler equation [48] over most ofthey E/Ee spectrum. In terms of the materialsradiation length X0, the differential cross section forbremsstrahlung radiation is:

d

dy =

A

NAX0

4

3+ C

1

y 1

+ y

, (8)

where C is a small material-dependent correction(AppendixA). Figure9 shows the integrated thick-ness of material upstream of the COT, in terms of ra-diation lengths, traversed by the reconstructed elec-tron tracks inW edata. The number of photonsemitted per layer is given by:

N= x

X0

4

3+ C

(y0 ln y0 1) +1

2(1 y0)2

,

(9)

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50

200

400

600

800

1000

1200

1400

1600

1800

2000

2200

2400

0x/X

NumberofElectrons/0.0

05

FIG. 9: The distribution of material upstream of the COTtraversed by reconstructed electron trajectories in Wedata events, in units of radiation lengths. The peaksat 0.08, 0.13, and 0.24 correspond respectivelyto trajectories outside the silicon detector (|z| >45 cm),within the silicon detector, and crossing silicon barrels(|z| 15, 45 cm). The mean of the distribution is 19%.

wherex is the thickness of the layer andy0 is a lowerthreshold introduced to avoid infrared divergences.We use y0 = 104 [49] and determine C = 0.0253using the silicon atomic number Z= 14.

For each layer of the silicon or COT material, weuse a Poisson distribution with mean Nto determinethe number of photons radiated in that layer. Foreach radiated photon, we calculate y from the spec-trum in Eq. (8). To correct for inaccuracies of thescreened Bethe-Heitler equation at the ends of the yspectrum, we apply a suppression factor ify 0.005or y 0.8.

For radiation of high-momentum photons (y 0.8), the approximation of complete screening of thenuclear electromagnetic field by the atomic electronsbreaks down. In this region, the full Bethe-Heitlerequation for incomplete screening[48] must be used.We implement this correction by removing generated

photons in the high-y region such that we match thereduced cross section from incomplete screening.

Two effects reduce the cross section for low-momentum photon radiation[50]: multiple scatteringand Compton scattering. Multiple Coulomb scatter-ing suppresses long-distance interactions, and the re-sultingLP M suppression [51] in low-momentum ra-diation can be expressed in terms of the Bethe-Heitlercross section[52]:

SLPM dLPM/dydBH/dy

=

ELPM

Ee

y

(1 y) , (10)

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where ELPM depends on the material. We useELPM= 72 TeV, appropriate for silicon, and apply

the suppression when SLPM

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0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

-210

-110

Photon E (GeV)

(E=100GeV)

-e+

e

(E)/

FIG. 11: The ratio of the Compton scattering cross sec-tion at a given photon energy to the pair-production crosssection atE= 100 GeV[54]. This ratio is applicable forphotons traversing silicon.

whereF(E) = 2.35e1.16E + 2.42e15.8E 5.21 0.151E, withEin GeV, and and are the initial-and final-state photons, respectively. We thus use thefollowing Compton scattering probability per layer:

P =RCom(7/9 C/6)x/X0. (16)

Energy Loss in Solenoid

After exiting the tracker electrons and photonstravel through the time-of-flight (TOF) system andthe solenoid. These systems have thicknesses of 10% and 85% of a radiation length, respec-tively. With this much material it becomes pro-hibitive to model individual radiative processes, andwe instead use a parametrized energy-loss model de-termined from a geant simulation [40]. The energy

loss is defined as the difference in energy of a sin-gle particle entering the TOF and the total energy ofparticles exiting the solenoid.

Figure12shows the mean energy loss as a functionof log10(pT/GeV) of the incoming particle for bothphotons and electrons. Electrons lose more energythan photons due to their ionization of the material.Since electrons withpT 400 MeV curve back to thecenter of the detector before exiting the solenoid, wedo not parametrize energy loss in this energy region.

The energy loss distribution at a given particlepTis reasonably described by an exponential. We usethis distribution, with a mean determined by Fig.12,

-2 -1.5 -1 -0.5 0 0.5 1 1.5 20

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

Photons

Electrons

Tp[10log (GeV)]

TpLossinTOFandSolenoid(GeV)

FIG. 12: The mean pT loss as a function of log10(pT/GeV) for electrons withpT >400 MeV and pho-tons traversing the time-of-flight system and solenoid.

to model the energy loss of a given particle passingthrough the TOF and solenoid.

Calorimeter Response and Fiduciality

The calorimeter simulation models the response of

the electromagnetic calorimeter as a function of eachparticles energy and position, and the fraction ofshower energy leaking into the hadronic calorimeter.

The electromagnetic calorimeter response, or theaverage measured energy divided by the true particleenergy entering the calorimeter, can depend on eachparticles energy. Possible sources of this dependenceare variations in light yield as a function of calorime-ter depth, attenuation in the light guide from thescintillator to the phototube, or leakage of showeringparticles into the hadronic calorimeter. The meanfractional energy leakage into the hadronic calorime-ter for particles exiting the tracker, determined us-

ing the geant calorimeter simulation, is shown as afunction of log10(pT/GeV) in Fig.13.

For a low-pTparticle exiting the tracker, the dis-tribution of energy loss into the hadronic calorimeteris adequately described by an exponential. For high-

pTparticles ( 10 GeV), the distribution has a peakat non-zero values of energy loss. In this energy re-gion we model the hadronic energy loss fluctuationswith the distributions shown in Fig. 13. Because anon-negligible fraction of electrons lose a significantamount of energy (510%) in the hadronic calorime-ter, it is important to model the energy loss spectrumin addition to the mean hadronic energy loss.

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0 0.5 1 1.5 2 2.50.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

Photons

Electrons

Tp[10log (GeV)]

Tp

Fractional

LeakageinHad.Ca

l.

0 0.02 0.04 0.06 0.08 0.11

10

210

310

Photons

Electrons

TpFractional Loss

ArbitraryScale

FIG. 13: The photon and electron pT leakage into thehadronic calorimeter. Top: The mean pT leakage as afunction of log10(pT/GeV). Bottom: The distributions ofpTleakage for high-pT (>10 GeV) photons and electrons.

To correct for any unaccounted dependence of theresponse on incoming particle energy, we use an em-pirical model of response that increases linearly withparticlepT:

REM(pT) = SE[1 + (pT/GeV 39)]. (17)We determine the slope parameter = [67(stat)] 105 using fits to the electron E/p distribution asa function of pT in W e and Z ee events(Section VI). The inclusive E/p distribution fromW e events is used to calibrate the absoluteresponse SE. Since electrons in this sample have amean pT of 39 GeV, the fitted values for SE and are uncorrelated. The parameter describes thenon-linearity of the calorimeter response.

Light attenuation in the scintillator results in non-uniform response as a function of distance from the

wavelength-shifting light guides. The attenuationfunction was measured using test beam data at con-

struction, and aging effects are measured insitu us-ing electrons from Wboson decays. The function isparametrized as a quadratic function of the CES xposition within a tower and corresponds to a reduc-tion in response of 10% at the edge of the tower.We simulate the light attenuation by reducing theenergy deposited by each particle according to thisfunction, evaluated at the particles CES x position.

To improve measurement resolution in data, wecorrect for attenuation effects by applying the inverseof the quadratic attenuation function to the measuredEM energy. We match this procedure in the simula-tion.

The EM calorimeter response drops rapidly as aparticle crosses the edge of the scintillator and intothe dead region between towers [30]. We take thecalorimeter to have zero response for any particlewith |CESx| >23.1 cm or |CESz| 1.06).

The final contribution to the electron cluster en-ergy comes from the underlying event [39] and ad-ditional pp interactions. As with muons, we mea-

sure this energy distribution in W boson data as afunction of u||, u, and the electron tower (Sec-tion VIIB). These measurements are incorporatedin the simulation.

IV. W BOSON SELECTION

The Wboson samples are collected with triggersrequiring at least one central (|| 1) lepton can-didate in the event. A narrow kinematic region isdefined for W boson selection: 30 GeV < lepton

pT < 55 GeV; 30 GeV < p/T < 55 GeV; 60 GeV

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< mT(l,p/T) < 100 GeV; and uT < 15 GeV. Thisselection results in low background while retaining

events with precise mW information. Additionalbackground rejection is achieved through event se-lection targeting the removal of Z boson decays toleptons. To minimize bias, lepton selection criteriaare required to have high efficiency or to be explic-itly modeled by our fast simulation.

A. W Selection

Muons are identified based on their reconstructedCOT track quality and production vertex, minimumionizing energy deposited in the calorimeter, and theconsistency of the track segments reconstructed inthe muon chambers with the COT tracks.

All charged lepton candidates from W and Z bo-son decay are required to have fully-fiducial cen-tral (|z0| 30 GeV. The two muonsmust have opposite charge and reconstruct to an in-variant mass in the 81 101 GeV range. The frac-tion of probe muons passing the additional W muoncandidate selection criteria is shown in Fig. 14as afunction of net recoil energy along the muon direction(u||). The observed dependence is parametrized as:

= a[1 + b(u||+ |u|||)], (19)wherea is a normalization factor that does not affectthemWmeasurement andb = [1.32 0.40(stat)]103. We vary b by3 in simulated data and fitfor mW. Assuming a linear variation ofmW with b,we derive uncertainties of mW = 1, 6, and 13 MeVfor themT, pT, andp/Tfits, respectively.

B. W e Selection

Electron identification uses information from the

COT track quality and production vertex, the match-ing of the track to calorimeter energy and position,and the longitudinal and lateral calorimeter energyprofiles.

An electron candidates COT track has the samefiduciality and hit usage requirements as a muon can-didate track, and utilizes the same beam-constrainedtrack fit. The track is required to havepT >18 GeV,a kinematic region where the trigger track-finding ef-ficiency has no pT dependence.

The clustering of showers in the CES produces anenergy-weighted position at the electron shower max-imum. We require the CES cluster to be well sepa-

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-15 -10 -5 0 5 10 150.9

0.92

0.94

0.96

0.98

1

Muons

(GeV)||u

Identificationefficiency/6GeV

FIG. 14: The muon identification efficiency as a functionof the recoil component in the direction of the muon (u||).

rated from the edges of the towers,|CESx| 9 cm. The cluster z position is com-pared to the extrapolated track z position, and thedifference is required to be less than 5 cm, consis-tent with the trigger requirement. The ratio of themeasured calorimeter energy to the track momentum,E/p, must be less than 2.

Electrons are differentiated from hadrons by their

high fraction of energy deposited in the electromag-netic calorimeter. The electrons EM energy is mea-sured in two neighboring towers in , while the energycollected in the hadronic calorimeter is measured inthree towers. The ratio,EHad/EEM, is required tobe less than 0.1. Only the EM calorimeter measure-ment is used to determine the electrons pT.

An electron shower will typically be confined toa single tower, with a small amount of energy flow-ing into the nearest tower in . We define an error-weighted difference between the observed and ex-pected energies in the two towers neighboring theelectron in the direction [56]:

Lshr = 0.14i

Eadji Eexpi0.142Eadji + (E

expi )

2

, (20)

whereEadji is the energy in a neighboring tower,Eexpi

is the expected energy contribution to that tower,Eexpi is the RMS of the expected energy, energiesare measured in GeV, and the sum is over the twoneighboring towers. We require Lshr < 0.3, consis-tent with the trigger criterion (Section IIIA 4).

The Z ee background is highly suppressed bytheuT 20 GeV and|d0| < 0.3cm extrapolates to a calorimeter region with reducedresponse (

|CES x

| > 22 cm or

|CES z

| < 6 cm),

and the tracks calorimeter isolation is < 0.1 (Sec-tionIV A). The full W e selection results in asample of 63,964 candidate events in (218.1 12.6)pb1 of integrated luminosity.

The track selection in the single-electron trigger(SectionIIIA4)results in an -dependent trigger ef-ficiency for reconstructed electrons (Fig. 15). Westudy this efficiency using Wevents selected with atrigger where the track requirements are replaced bya p/Tthreshold. The efficiency decreases as|| de-creases because the reduced path length reduces theionization charge collected by each wire, thus reduc-ing the single hit efficiency. There is an additional

decrease in efficiency due to the dead region at |z| 2mm. Electrons crossing this region at track|| = 0are not included in the efficiency plot, since we onlymeasure electrons with |CES z | >9 cm. Thus, at|| = 0 there is no inefficiency due to the dead COTregion, and the measured efficiency increases.

We measure the u|| dependence of the electronidentification efficiency (Fig. 16) using Z eeevents, selected with one electron passing theWelec-tron candidate criteria and a second probe elec-tron identified as an EM energy cluster with pT >30GeV, an associated track with pT > 18 GeV, andE/p

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-15 -10 -5 0 5 10 150.95

0.96

0.97

0.98

0.99

1

Electrons

(GeV)||u

Identificationefficiency/6GeV

FIG. 16: The electron identification efficiency measuredin Z ee data as a function of the recoil componentin the direction of the electron (u||). Background is sub-tracted using the number of like-charge lepton events ob-served at a given u||. The E/p < 2 requirement is notincluded in this efficiency measurement.

an E/p requirement, this cut is not included in theefficiency measurement. We instead study the unbi-ased E/p < 2 efficiency by recalculating E and u||for towers separated in from the identified electronin W

e events, and find no significant u

|| de-

pendence in this efficiency. In the simulation we useb= 0 0.54 103, obtained by fitting the measuredefficiencies to the function in Eqn.19.

We vary b by3 in pseudoexperiments and as-sume linear variation ofmW with b to derive uncer-tainties ofmW = 3, 5, and 16 MeV for the mT, pT,andp/Tfits, respectively. Sincebis measured with dif-ferent data samples for the electron and muon chan-nels, there is no correlation between the correspond-ing systematic uncertainties.

V. TRACK MOMENTUM MEASUREMENT

Muon momenta are determined from helical fits totracks reconstructed using COT information. Themomentum resolution of prompt muons is improvedby constraining the helix to originate from the trans-verse beam position. A given muons transverse mo-mentum is determined by the Lorentz equation,

mv2/R = evB,

pT = eB/(2|c|),(21)

where B is the magnetic field, R is the radius ofcurvature, c q/(2R) is the curvature of the he-

lix, and q is the muon charge. The a priori mo-mentum scale is determined by the measurements of

the magnetic field and the radius of the tracker. AtCDF, eB/2 = 2.11593 103 GeV/cm, where B ismeasured using an NMR probe at a COT endplate.Measurements of the local field nonuniformities andtracker geometry were performed during constructionand installation and are used to determine the posi-tions of individual track hits. We find these measure-ments provide an a priorimomentum scale accuracyof 0.15%.

We refine the momentum scale calibration withdata. Using reconstructed cosmic ray muon tracks,we align the relative positions of the tracker wires.Track-level corrections derived from W e datareduce relative curvature bias between positive andnegative particles. Finally, we perform an abso-lute calibration of the momentum scale using high-statistics data samples ofJ/, , and Z boson de-cays to muons. The final calibration is applied as arelative momentum correction p/p to theWbosondata and has an accuracy of 0.02%.

A. COT Alignment

The COT contains 30,240 sense wires for measur-ing the positions of charged particles passing through

the detector. The position measurements rely on anaccurate knowledge of the wire positions throughoutthe chamber. We determine these positions using acombination of alignment survey, computer model-ing, and cosmic-ray muon data. Any remaining bi-ases in track parameter measurements are studiedwith J/ and W e data, from which fi-nal track-level corrections are derived.

After construction of the COT endplates, the po-sition of each 12-wire cell was measured with an ac-curacy of13m using a coordinate measuring ma-chine. The effect of the load of the wire plane andfield sheets was modeled with a finite element analysis

(FEA) and found to cause an endplate bend towardsz = 0 cm, with the maximum bend of 6 mm inthe fifth superlayer[29]. An equivalent load was ap-plied to the detector and further measurements foundthe FEA to be accurate to within 20%. The FEAresults were scaled to match the measurements, andthe positions determined from the FEA were set asthe directly-determined cell positions.

While each cell position determines the average po-sitions of its 12 sense wires within the chamber, sev-eral effects create a non-linear wire shape as a func-tion of z. Gravity has the most significant effect,causing each wire to sag 260 m in y at z = 0

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Sense wires

R

r

FIG. 17: The definitions of the local tilt () and az-imuthal shift (R) alignment corrections applied toeach COT cell.

cm. Electrostatic deflection towards the nearest fieldsheet occurs for cells where the sense wire is not cen-tered between the field sheets. By construction, thewires are slightly offset within a cell; in addition, thegravitational sag of the field sheets is larger than thatof the sense wires, resulting in an electrostatic deflec-

tion that partially counteracts the sag of the sensewires. Combined, the electrostatic effects cause a -dependent wire shift that has a maximum of 74 mat = 145o and z = 0 cm. The gravitational andelectrostatic effects were combined to determine thebest a prioriestimate of the wire shapes.

Starting from the predicted cell and wire positions,we develop in situ corrections based on cosmic-raymuon data taken during pp crossings with the sin-gle muon trigger. The data are selected by requiringexactly two reconstructed tracks in the event, elim-inating effects from overlapping hits from collision-induced particles. Since the two tracks on oppo-

site sides of the COT result from a single cosmic-raymuon, we refit both tracks to a single helix and de-termine hit residuals with respect to this helix [57].For each cell, we use the residuals to determine atilt correction about its center, and a shift correctionalong the global azimuth (Fig. 17). We show thetilt and shift corrections for the inner superlayer ofthe west endplate in Fig. 18, after removing globalcorrections. We apply these corrections to each cellof each superlayer in each endplate. In addition, wemeasure a relative east-west shift and include it ineach cells correction.

We combine the cell-based corrections with wire-

0 20 40 60 80 100 120 140 160-0.003

-0.002

-0.001

0

0.001

0.002

0.003

Cell Number

/C

ell

0 20 40 60 80 100 120 140 160

-0.04

-0.02

0

0.02

0.04

Cell Number

(cm)/Cell

R

FIG. 18: The local tilt (top) and azimuthal shift (bottom)alignment corrections applied to each cell of the innersuperlayer of the west endplate. Not shown are a global0.0021 tilt correction and a small global rotation and shiftof the COT that does not affect track measurements.

based corrections for the shapes of the wires betweenthe endplates. We measure these corrections as func-tions ofz and radius R using the differences in themeasuredd0 and curvature parameters for the helixfits on opposite sides of the COT for a cosmic ray

muon. The corrections are applied as additional off-sets of the wires at z = 0 cm, with a parabolicwire shape as a function ofz. The corrections includea radial dependence,

= 160 + 380(R/140) 380(R/140)2, (22)where R is measured in cm and in m. Figure19shows the gravitational and electrostatic shifts ofa wire as a function of z at = , as well as thedata-based correction at R = 130 cm (the outer su-perlayer).

The cell- and wire-based corrections are imple-mented for the track-finding and fitting stage, and re-

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0 100

-200

-100

0

100

Gravitational calculationWith electrostatics calculation

With cosmic ray data correction

z (cm)

=

R = 130 cm

m)

y(

FIG. 19: The net wire shift iny as a function ofz fromgravitational sag only (solid), including electrostatic ef-fects (dashed), and including data-based corrections fromEq. (22) (dotted). The shift is shown at = andR= 130 cm.

duce the measured hit resolution for high-momentummuons from 180m to 140m. Final track-basedcorrections are applied to the measured track cur-vature, which is inversely related to the transversemomentum [Eq. (21)]. Expanding the measured cur-

vature c as a function of the true curvature ct in aTaylor series around zero,

c= 1+ (1 + 2)ct+ 3c2t + 4c

3t + ..., (23)

the terms even inctcause biases in positive tracks rel-ative to negative tracks, which tend to cancel whenthe two are averaged. The term linear in ct scales thetrue curvature and is determined by the momentumcalibration. The4c

3t term is the first to directly af-

fect mass measurements and is suppressed by the c3tfactor at low curvature (high momentum).

Corrections for high-momentum tracks from Wand Zdecay particles are determined using the dif-

ference in E /p for e+

and e

fromWdecays, whichshould be zero in the absence of misalignments. Thisdifference can be used to constrain 1, the first termin the Taylor expansion. Figure20 shows the differ-ences in E /p as functions of cot and , before andafter corrections of the following form:

c = a0+ a1cot + a2cot2 +

b1 sin( + 0.1) + b3sin(3 + 0.5).(24)

The terms can be interpreted as arising from thefollowing physical effects: a relative rotation of theouter edge to the inner edge of each endplate (a0);

-1 -0.5 0 0.5 1-0.02

-0.01

0

0.01

0.02

COT cell and wire alignment

With track-level corrections

cot

E/p(positrons-electrons)

0 1 2 3 4 5 6

-0.04

-0.02

0

0.02

0.04

E/p(positrons-electrons)

FIG. 20: The difference between e+ and e E/p as afunction of cot (top) and (bottom) before (triangles)and after (diamonds) track-level corrections.

a relative rotation of the east and west endplates(a1cot ); and a mismeasurement of the beam posi-tion (b1sin( + 0.1)). The measured values of the pa-rametersa0, a1, a2, b1,and b3, are shown in TableIII.

Varyinga1 by3 in pseudoexperiments and as-suming linear variation of the momentum scale witha1

, we find thea1

uncertainty results in a relative mo-mentum scale uncertainty of0.07103 forWandZ boson mass measurements. The other parameteruncertainties, as well as residual higher-order terms,have a negligible impact on the momentum scale forthemWmeasurement.

B. J/ Calibration

With a measured BR of 16.3+1.41.3 nb [28],J/ mesons are the Tevatrons most prolific sourceof resonant decays to muon pairs. In addition to

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Parameter Value (107 cm1)a0

0.66

0.17

a1 1.6 0.3a2 2.1 0.5b1 2.1 0.2b3 5.7 1.7

TABLE III: The parameters used to correct the trackcurvature of electrons and muons from W and Z bosondecays. The values and statistical uncertainties are deter-mined from fits to the E/p difference between positronsand electrons.

its high statistics, the J/s precisely known mass(mJ/ = 3096.88 0.04 MeV[58]) and narrow width(J/ = 0.0934 0.0021 MeV [11]) make it a keycomponent of the track momentum calibration. Weperform measurements of theJ/ mass as a functionof mean inverse muon pT to determine a momentumscale correction and extrapolate to the high-pT regionrelevant for W and Zboson decays.

1. Data Sample

The J/ data sample is collected with a Level 1

trigger requiring onepT >1.5 GeV XFT track with amatching CMU track segment, and a second pT >1.5(2) GeV XFT track with a matching CMU (CMX)segment. At Level 3, the two corresponding COTtracks must have opposite charge and consistent zvertex positions (|z0| < 5 cm), and must form aninvariant mass between 2.7 and 4 GeV. The resolutionon the invariant mass measurement degrades at hightrack momentum, so to avoid trigger bias the massrange is extended to 2 GeV < m < 5 GeV whenthepTof the muon pair p

T is greater than 9 GeV.

Candidate events are selected offline by requiringtwo COT tracks, each with pT > 2 GeV,|d0| < 0.3cm, and 7 hits on each of the eight superlayers.The tracks must originate from a common vertex(|z0| < 3 cm) and form an invariant mass in therange (2.95, 3.21) GeV.

A significant fraction ( 20%) of the J/ mesonsin our data sample result from decays ofB hadrons,which have an average proper decay length of 0.5mm. The muons from theJ/ decay can thus orig-inate outside the beam radius. Therefore, no beamconstraint is applied in the COT track fit of muoncandidates from J/ decays.

The total sample consists of 606,701 J/ candi-dates in (194.1 11.3) pb1 of integrated luminosity.

2. Monte Carlo Generation

We use pythia[59] to generateJ/ events,from which templates are constructed to fit the datafor the momentum scale. The shape of them dis-tribution from J/ decays is dominated by the pT-dependent detector resolution. We therefore model

the pJ/T distribution as well as the pT and relative

pTof the muons in a J / decay. To obtain an ade-quate model, we empirically tune the generated J/kinematics to describe the relevant data distributionsfor theJ/ mass fits.

To tune the pJ/T distribution, we boost the J/

momentum by changing its rapidity (yJ/) along itsdirection of motion pJ/. In 50% of the generatedevents we multiply yJ/ by 1.215, and in the other50% we multiply it by 1.535. The decay angle

in the J/ rest frame relative to pJ/ is tuned bymultiplying cot by 1.3. After tuning, the simula-tion matches the relevant background-corrected datadistributions, as shown in Fig. 21.

The pythia event generator does not include en-ergy loss due to final-state photon radiation from themuons in J/ decays. To simulate this effect, wescale each muons momentum by a factor x deter-mined from the following leading-log probability dis-tribution for soft photon radiation [59,60]:

f(x) = (1 x)1

, (25)

with

=EM

[ln(Q2/m2) 1] (26)

andQ2 =m2J/.

3. Momentum Scale Measurement

The momentum scale is calibrated using J/ de-cays by fitting the dimuon mass as a function of mean

inverse pTof the two muons, and then extrapolatingto high pT (

p1T

0 GeV1). This procedure re-sults in a track momentum calibration accuracy of0.025%.

The momentum scale calibration requires an accu-rate modeling of the muon ionization energy loss inthe tracker. Each muon passing through the siliconand COT detectors loses on average 9 MeV at normalincidence. The combined effect on the reconstructedm is about 0.6% ofmJ/, a factor of 20 largerthan our total uncertainty. Since the ionization en-ergy lossEIvaries only logarithmically with pT(Sec-tionIIIB 1), the relative effect on the reconstructed

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0 2 4 6 8 10 12 140

20000

40000

60000 J /

Tp (GeV)

NumberofEvents/0.5GeV

-0.4 -0.2 0 0.2 0.4

20

40

60

80

100

120

140

310

Tq/p )-1(GeV

-1

NumberofEvents/0.1GeV

FIG. 21: TheJ/

data (points) and tuned simula-tion (histogram) distributions of pT (top) and

Pq/pT

(bottom). TheP

q/pT is equal to the sum of thetrack curvatures of muons from a J/ decay, divided by2.11593 103.

mass is:

m

m =

E+

I

2p+

T

+ E

I

2p

T

EIp1T

. (27)

Thus, in a linear fit of m/mas a function of mean

inverse pT, a non-zero slope approximately corre-sponds to EI. Since we model the ionization energyloss based on the known detector material, this slopeshould be zero. We however find that we need toscale down the ionization energy loss from the detec-tor parametrization (SectionIIIB 1) by 6% to achievea zero slope. We show the result of this tuning inFig.22, replacing m/mon theyaxis with the rel-ative momentum correction p/pto be applied to thedata in order to measure mJ/ = 3096.88 MeV. The

tuning is based on ap1T

region of (0.1, 0.5) GeV1,

divided into eight bins. We find a scale correction ofp/p= [1.64 0.06(stat)] 103 from a linear fit

0 0.2 0.4 0.6-0.003

-0.002

-0.001

0

Tp

p/

p

J

< 1 / )-1

> (GeV

-3100.06)Scale correction = (-1.64

GeV-3

100.17)Energy loss = (-0.05

data/

FIG. 22: The fractional momentum correction for data asa function of the mean inverse momentum of the muonsfromJ/ decays. In a linear fit, the intercept correspondsto the scale correction relevant forWandZboson decays,and the slope corresponds to the remaining unmodeledionization energy loss after material tuning. The uncer-tainties are statistical only.

to p/p as a function ofp1T

.

Each p/p value in Fig. 22 is extracted via abinned likelihood fit to the mdistribution for eachp1

T

bin. Since the mass resolution varies signifi-cantly with

p1T

, the fit ranges are adjusted from

3.08 0.13 GeV for p1T = (0.1, 0.15) GeV1 to3.08 0.08 GeV for p1T = (0.45, 0.5) GeV1. Thebackground is modeled as a linear function ofm,with normalization and slope determined from upperand lower sideband regions whose combined widthis equal to that of the mass fit window. The re-sults of the fits in the

p1T

= (0.15, 0.2) GeV1

andp1T

= (0.25, 0.3) GeV1 ranges are shown in

Fig. 23.The J/ momentum calibration includes correc-

tions to the curvature c derived from the measured

dimuon mass as a function of cot between thepositive and negative muons from the J/ decay. Bi-ases linear in cot are removed with a curvaturecorrection linear in cot :

c = [(7 1) 107 cm1]cot , (28)

where the uncertainty is statistical only. Biasesquadratic in cot are removed with the followingcorrection to the absolute length scale of the COTalong thez axis (statistical uncertainty only):

cot = [(3.75 1.00) 104]cot . (29)

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3 3.2 3.40

1000

2000

/dof = 17 / 222

m

Tp >-1-1 10 statis-tical variation, results in a0.03 103 change inp/p. We include this in our systematic uncertaintyestimate.

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A 0.03103 uncertainty on p/p from the back-ground model is determined by changing its linear de-

pendence on m to a constant. Finally, the world-average J/ mass value used in this measurementcontributes 0.01103 to the uncertainty on p/p.

The final momentum scale correction derived fromJ/ data is:

p/p= (1.64 0.25) 103. (30)

C. Calibration

Thebbresonance provides a complementary mo-mentum scale calibration tool to the J/. Its pre-cisely measured mass m = (9460.30 0.26) MeV[11] is three times larger than that of the J/, soan momentum scale calibration is less sensitive tothe material and energy loss model than that of theJ/. Because thebbresonances are the highest massmesons, long-lived hadrons do not decay to the andthe muons from decay effectively originate fromthe collision point. We improve the accuracy of themuon measurements by constraining their tracks tothe beam position, which is the same procedure ap-plied to the W andZdecay lepton tracks.

The data sample is based on the same Level1 trigger as the J/ sample (Section V B 1). The

Level 3 requirements are: one reconstructed trackwith pT > 4 GeV and matching CMU and CMPtrack segments (CMUP); a second track with oppo-site charge to the first, pT >3 GeV, and a matchingCMU or CMX track segment; and a reconstructedmass of the two tracks between 8 and 12 GeV. Of-fline, thepTthresholds are increased to 4.2 (3.2) GeVfor the track with a CMUP (CMU or CMX) track seg-ment, and each track must have |d0|

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Source J/ (103) (103) Common (103)QED and energy loss model 0.20 0.13 0.13

Magnetic field nonuniformities 0.10 0.12 0.10

Beam constraint bias N/A 0.06 0

Ionizing material scale 0.06 0.03 0.03

COT alignment corrections 0.05 0.03 0.03

Fit range 0.05 0.02 0.02

Trigger efficiency 0.04 0.02 0.02

Resolution model 0.03 0.03 0.03

Background model 0.03 0.02 0.02

World-average mass value 0.01 0.03 0

Statistical 0.01 0.06 0

Total 0.25 0.21 0.17

TABLE IV: Uncertainties on the momentum scale correction derived from the J/ and mass measurements.

model cannot be tested with the 2 of the mass fit. Instead, we changeQ in the photon radia-tion probability by the amount estimated for theJ/systematic uncertainty (SectionV B 4). We find thatthis variation affects p/pby0.13 103 in the calibration.

The final result of the calibration is:

p/p= (1.44 0.21) 103. (31)We have verified that this result has no time de-pendence, at the level of the statistical precision of0.13 103. When combined with the momentumscale correction from the J/ calibration, we obtain:

p/p= (1.50 0.19) 103. (32)

D. Z Calibration

Given the precise momentum scale calibration fromtheJ/ and decays, we measure theZboson massand compare it to the world-average value mZ =

(91187.6 2.1) MeV [11]. We then use the world-averagemZto derive an additional p/pcalibrationand combine it with that of the J/ and decays.

The systematic uncertainties of the mZ measure-ment are correlated with those of the mW mea-surement, so a momentum scale calibration with Zbosons can reduce systematic uncertainties on themW measurement. However, the statistical uncer-tainty from theZ sample is significantly largerthan the calibration uncertainty fromJ/and de-cays. Thus, the main purpose of the mZ measure-ment is to confirm the momentum scale calibrationand test our systematic uncertainty estimates.

TheZboson data sample is selected using the samesingle-muon trigger and offline muon selection as fortheWboson sample (SectionsIII A4andIV A), withthe exception that we remove the requirement of atrack segment in a muon detector for one of themuons from the Z boson decay. Removing this re-quirement significantly increases detector acceptancewhile negligibly affecting background. Zboson can-didates are defined by 66 GeV < m < 116 GeV,

pT < 30 GeV,|t0(, )| < 3 ns, and oppositely

charged muons. A muon trackst0 is defined as thetime between the pp bunch crossing and the muonsproduction, and should be (0 1) ns for Z production and decay. The track t0 is measured us-ing the time informati