Combination of the top-quark mass measurements from the Tevatron collider

T. Aaltonen,12 V.M. Abazov,48 B. Abbott,112 B. S. Acharya,31 M. Adams,78 T. Adams,74 G.D. Alexeev,48 G. Alkhazov,52

A. Alton,96,b B. Álvarez González,57,c G. Alverson,92 S. Amerio,35a D. Amidei,96 A. Anastassov,76,d A. Annovi,34

J. Antos,53 G. Apollinari,76 J. A. Appel,76 T. Arisawa,41 A. Artikov,48 J. Asaadi,119 W. Ashmanskas,76 A. Askew,74

S. Atkins,89 B. Auerbach,72 K. Augsten,9 A. Aurisano,119 C. Avila,7 F. Azfar,66 F. Badaud,13 W. Badgett,76 T. Bae,43

L. Bagby,76 B. Baldin,76 D. V. Bandurin,74 S. Banerjee,31 A. Barbaro-Galtieri,68 E. Barberis,92 P. Baringer,87

V. E. Barnes,85 B.A. Barnett,90 P. Barria,36a,36c J. F. Bartlett,76 P. Bartos,53 U. Bassler,18 M. Bauce,35a,35b V. Bazterra,78

A. Bean,87 F. Bedeschi,36a M. Begalli,2 S. Behari,90 L. Bellantoni,76 G. Bellettini,36a,36b J. Bellinger,125 D. Benjamin,109

A. Beretvas,76 S. B. Beri,29 G. Bernardi,17 R. Bernhard,22 I. Bertram,61 M. Besançon,18 R. Beuselinck,63 P. C. Bhat,76

S. Bhatia,99 V. Bhatnagar,29 A. Bhatti,105 D. Bisello,35a,35b I. Bizjak,64 K. R. Bland,122 G. Blazey,79 S. Blessing,74

K. Bloom,100 B. Blumenfeld,90 A. Bocci,109 A. Bodek,106 A. Boehnlein,76 D. Boline,107 E. E. Boos,50 G. Borissov,61

D. Bortoletto,85 T. Bose,91 J. Boudreau,116 A. Boveia,77 A. Brandt,118 O. Brandt,23 L. Brigliadori,33a,33b R. Brock,98

C. Bromberg,98 A. Bross,76 D. Brown,17 J. Brown,17 E. Brucken,12 X. B. Bu,76 J. Budagov,48 H. S. Budd,106 M. Buehler,76

V. Buescher,25 V. Bunichev,50 S. Burdin,61,e K. Burkett,76 G. Busetto,35a,35b P. Bussey,60 C. P. Buszello,58 A. Buzatu,4

A. Calamba,115 C. Calancha,56 E. Camacho-Pérez,45 S. Camarda,54 M. Campanelli,64 M. Campbell,96 F. Canelli,77

B. Carls,81 D. Carlsmith,125 R. Carosi,36a S. Carrillo,73,f S. Carron,76 B. Casal,57,g M. Casarsa,38a B. C. K. Casey,76

H. Castilla-Valdez,45 A. Castro,33a,33b P. Catastini,93 S. Caughron,98 D. Cauz,38a V. Cavaliere,81 M. Cavalli-Sforza,54

A. Cerri,68,h L. Cerrito,64,i S. Chakrabarti,107 D. Chakraborty,79 K.M. Chan,84 A. Chandra,121 E. Chapon,18 G. Chen,87

Y. C. Chen,6 M. Chertok,69 S. Chevalier-Théry,18 G. Chiarelli,36a G. Chlachidze,76 F. Chlebana,76 D. K. Cho,117 K. Cho,43

S.W. Cho,44 S. Choi,44 D. Chokheli,48 B. Choudhary,30 W.H. Chung,125 Y. S. Chung,106 S. Cihangir,76 M.A. Ciocci,36a,36c

D. Claes,100 A. Clark,59 C. Clarke,97 J. Clutter,87 G. Compostella,35a,35b M. E. Convery,76 J. Conway,69 M. Cooke,76

W. E. Cooper,76 M. Corbo,76 M. Corcoran,121 M. Cordelli,34 F. Couderc,18 M.-C. Cousinou,15 C.A. Cox,69 D. J. Cox,69

F. Crescioli,36a,36b A. Croc,18 J. Cuevas,57,c R. Culbertson,76 D. Cutts,117 D. Dagenhart,76 A. Das,67 N. d’Ascenzo,76,j

M. Datta,76 G. Davies,63 P. de Barbaro,106 S. J. de Jong,46,47 E. De La Cruz-Burelo,45 F. Déliot,18 M. Dell’Orso,36a,36b

R. Demina,106 L. Demortier,105 M. Deninno,33a D. Denisov,76 S. P. Denisov,51 M. d’Errico,35a,35b S. Desai,76 C. Deterre,18

K. DeVaughan,100 F. Devoto,12 A. Di Canto,36a,36b B. Di Ruzza,76 H. T. Diehl,76 M. Diesburg,76 P. F. Ding,65

J. R. Dittmann,122 A. Dominguez,100 S. Donati,36a,36b P. Dong,76 M. D’Onofrio,62 M. Dorigo,38a T. Dorigo,35a A. Dubey,30

L. V. Dudko,50 D. Duggan,101 A. Duperrin,15 S. Dutt,29 A. Dyshkant,79 M. Eads,100 K. Ebina,41 D. Edmunds,98

A. Elagin,119 J. Ellison,71 V.D. Elvira,76 Y. Enari,17 A. Eppig,96 R. Erbacher,69 S. Errede,81 N. Ershaidat,76,k R. Eusebi,119

H. Evans,82 A. Evdokimov,108 V.N. Evdokimov,51 G. Facini,92 S. Farrington,66 M. Feindt,24 L. Feng,79 T. Ferbel,106

J. P. Fernandez,56 F. Fiedler,25 R. Field,73 F. Filthaut,46,47 W. Fisher,98 H. E. Fisk,76 G. Flanagan,76,l R. Forrest,69

M. Fortner,79 H. Fox,61 M. J. Frank,122 M. Franklin,93 J. C. Freeman,76 S. Fuess,76 Y. Funakoshi,41 I. Furic,73

M. Gallinaro,105 J. E. Garcia,59 A. Garcia-Bellido,106 J. A. Garcı́a-González,45 G.A. Garcı́a-Guerra,45,m A. F. Garfinkel,85

P. Garosi,36a,36c V. Gavrilov,49 P. Gay,13 W. Geng,15,98 D. Gerbaudo,102 C. E. Gerber,78 H. Gerberich,81 E. Gerchtein,76

Y. Gershtein,101 S. Giagu,37a V. Giakoumopoulou,28 P. Giannetti,36a K. Gibson,116 C.M. Ginsburg,76 G. Ginther,76,106

N. Giokaris,28 P. Giromini,34 G. Giurgiu,90 V. Glagolev,48 D. Glenzinski,76 M. Gold,103 D. Goldin,119 N. Goldschmidt,73

A. Golossanov,76 G. Golovanov,48 G. Gomez,57 G. Gomez-Ceballos,94 M. Goncharov,94 O. González,56 I. Gorelov,103

A. T. Goshaw,109 K. Goulianos,105 A. Goussiou,124 P. D. Grannis,107 S. Greder,19 H. Greenlee,76 G. Grenier,20

S. Grinstein,54 Ph. Gris,13 J.-F. Grivaz,16 A. Grohsjean,18,n C. Grosso-Pilcher,77 R. C. Group,123,76 S. Grünendahl,76

M.W. Grünewald,32 T. Guillemin,16 J. Guimaraes da Costa,93 G. Gutierrez,76 P. Gutierrez,112 S. Hagopian,74 S. R. Hahn,76

J. Haley,92 E. Halkiadakis,101 A. Hamaguchi,40 J. Y. Han,106 L. Han,5 F. Happacher,34 K. Hara,42 K. Harder,65 D. Hare,101

M. Hare,95 A. Harel,106 R. F. Harr,97 K. Hatakeyama,122 J.M. Hauptman,86 C. Hays,66 J. Hays,63 T. Head,65 T. Hebbeker,21

M. Heck,24 D. Hedin,79 H. Hegab,113 J. Heinrich,114 A. P. Heinson,71 U. Heintz,117 C. Hensel,23 I. Heredia-De La Cruz,45

M. Herndon,125 K. Herner,96 G. Hesketh,65,o S. Hewamanage,122 M.D. Hildreth,84 R. Hirosky,123 T. Hoang,74

J. D. Hobbs,107 A. Hocker,76 B. Hoeneisen,11 J. Hogan,121 M. Hohlfeld,25 W. Hopkins,76,p D. Horn,24 S. Hou,6

I. Howley,118 Z. Hubacek,9,18 R. E. Hughes,110 M. Hurwitz,77 U. Husemann,72 N. Hussain,4 M. Hussein,98 J. Huston,98

V. Hynek,9 I. Iashvili,104 Y. Ilchenko,120 R. Illingworth,76 G. Introzzi,36a M. Iori,37a,37b A. S. Ito,76 A. Ivanov,69,q

S. Jabeen,117 M. Jaffré,16 E. James,76 D. Jang,115 A. Jayasinghe,112 B. Jayatilaka,109 E. J. Jeon,43 M. S. Jeong,44 R. Jesik,63

S. Jindariani,76 K. Johns,67 E. Johnson,98 M. Johnson,76 A. Jonckheere,76 M. Jones,85 P. Jonsson,63 K. K. Joo,43 J. Joshi,71

S. Y. Jun,115 A.W. Jung,76 T. R. Junk,76 A. Juste,55 K. Kaadze,88 E. Kajfasz,15 T. Kamon,43,119 P. E. Karchin,97

PHYSICAL REVIEW D 86, 092003 (2012)

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D. Karmanov,50 A. Kasmi,122 P. A. Kasper,76 Y. Kato,40,r I. Katsanos,100 R. Kehoe,120 S. Kermiche,15 W. Ketchum,77

J. Keung,114 N. Khalatyan,76 A. Khanov,113 A. Kharchilava,104 Y.N. Kharzheev,48 V. Khotilovich,119 B. Kilminster,76

D.H. Kim,43 H. S. Kim,43 J. E. Kim,43 M. J. Kim,34 S. B. Kim,43 S. H. Kim,42 Y. J. Kim,43 Y.K. Kim,77 N. Kimura,41

M. Kirby,76 I. Kiselevich,49 S. Klimenko,73 K. Knoepfel,76 J.M. Kohli,29 K. Kondo,41,a D. J. Kong,43 J. Konigsberg,73

A. V. Kotwal,109 A. V. Kozelov,51 J. Kraus,99 M. Kreps,24 J. Kroll,114 D. Krop,77 M. Kruse,109 V. Krutelyov,119,s T. Kuhr,24

S. Kulikov,51 A. Kumar,104 A. Kupco,10 M. Kurata,42 T. Kurča,20 V. A. Kuzmin,50 S. Kwang,77 A. T. Laasanen,85

S. Lami,36a S. Lammel,76 S. Lammers,82 M. Lancaster,64 R. L. Lander,69 G. Landsberg,117 K. Lannon,110,t A. Lath,101

G. Latino,36a,36c P. Lebrun,20 T. LeCompte,75 E. Lee,119 H. S. Lee,44 H. S. Lee,77,u J. S. Lee,43 S.W. Lee,86 W.M. Lee,76

S.W. Lee,119,v X. Lei,67 J. Lellouch,17 S. Leo,36a,36b S. Leone,36a J. D. Lewis,76 H. Li,14 L. Li,71 Q. Z. Li,76 J. K. Lim,44

A. Limosani,109,w C.-J. Lin,68 D. Lincoln,76 M. Lindgren,76 J. Linnemann,98 V.V. Lipaev,51 E. Lipeles,114 R. Lipton,76

A. Lister,59 D.O. Litvintsev,76 C. Liu,116 H. Liu,120 H. Liu,123 Q. Liu,85 T. Liu,76 Y. Liu,5 A. Lobodenko,52 S. Lockwitz,72

A. Loginov,72 M. Lokajicek,10 R. Lopes de Sa,107 H. J. Lubatti,124 D. Lucchesi,35a,35b J. Lueck,24 P. Lujan,68 P. Lukens,76

R. Luna-Garcia,45,x G. Lungu,105 A. L. Lyon,76 J. Lys,68 R. Lysak,53,y A.K.A. Maciel,1 R. Madar,18 R. Madrak,76

K. Maeshima,76 P. Maestro,36a,36c R. Magaña-Villalba,45 S. Malik,105 S. Malik,100 V. L. Malyshev,48 G. Manca,62,z

A. Manousakis-Katsikakis,28 Y. Maravin,88 F. Margaroli,37a C. Marino,24 M. Martı́nez,54 J. Martı́nez-Ortega,45

P. Mastrandrea,37a K. Matera,81 M. E. Mattson,97 A. Mazzacane,76 P. Mazzanti,33a R. McCarthy,107 K. S. McFarland,106

C. L. McGivern,65 P. McIntyre,119 R. McNulty,62,aa A. Mehta,62 P. Mehtala,12 M.M. Meijer,46,47 A. Melnitchouk,99

D. Menezes,79 P. G. Mercadante,3 M. Merkin,50 C. Mesropian,105 A. Meyer,21 J. Meyer,23 T. Miao,76 F. Miconi,19

D. Mietlicki,96 A. Mitra,6 H. Miyake,42 S. Moed,76 N. Moggi,33a N. K. Mondal,31 M.N. Mondragon,76,f C. S. Moon,43

R. Moore,76 M. J. Morello,36a,36d J. Morlock,24 P. Movilla Fernandez,76 A. Mukherjee,76 M. Mulhearn,123 Th. Muller,24

P. Murat,76 M. Mussini,33a,33b J. Nachtman,76,bb Y. Nagai,42 J. Naganoma,41 E. Nagy,15 M. Naimuddin,30 I. Nakano,39

A. Napier,95 M. Narain,117 R. Nayyar,67 H. A. Neal,96 J. P. Negret,7 J. Nett,119 C. Neu,123 M. S. Neubauer,81 P. Neustroev,52

J. Nielsen,68,cc L. Nodulman,75 S. Y. Noh,43 O. Norniella,81 T. Nunnemann,26 L. Oakes,66 S. H. Oh,109 Y.D. Oh,43

I. Oksuzian,123 T. Okusawa,40 R. Orava,12 J. Orduna,121 L. Ortolan,54 N. Osman,15 J. Osta,84 M. Padilla,71

S. Pagan Griso,35a,35b C. Pagliarone,38a A. Pal,118 E. Palencia,57,h V. Papadimitriou,76 A.A. Paramonov,75 N. Parashar,83

V. Parihar,117 S. K. Park,44 R. Partridge,117,dd N. Parua,82 J. Patrick,76 A. Patwa,108 G. Pauletta,38a,38b M. Paulini,115

C. Paus,94 D. E. Pellett,69 B. Penning,76 A. Penzo,38a M. Perfilov,50 Y. Peters,65 K. Petridis,65 G. Petrillo,106 P. Pétroff,16

T. J. Phillips,109 G. Piacentino,36a E. Pianori,114 J. Pilot,110 K. Pitts,81 C. Plager,70 M.-A. Pleier,108

P. L.M. Podesta-Lerma,45,ee V.M. Podstavkov,76 L. Pondrom,125 A.V. Popov,51 S. Poprocki,76,p K. Potamianos,85

A. Pranko,68 M. Prewitt,121 D. Price,82 N. Prokopenko,51 F. Prokoshin,48,ff F. Ptohos,34,gg G. Punzi,36a,36b J. Qian,96

A. Quadt,23 B. Quinn,99 A. Rahaman,116 V. Ramakrishnan,125 M. S. Rangel,1 K. Ranjan,30 N. Ranjan,85 P. N. Ratoff,61

I. Razumov,51 I. Redondo,56 P. Renkel,120 P. Renton,66 M. Rescigno,37a T. Riddick,64 F. Rimondi,33a,33b I. Ripp-Baudot,19

L. Ristori,36a,76 F. Rizatdinova,113 A. Robson,60 T. Rodrigo,57 T. Rodriguez,114 E. Rogers,81 S. Rolli,95,hh M. Rominsky,76

R. Roser,76 A. Ross,61 C. Royon,18 P. Rubinov,76 R. Ruchti,84 F. Ruffini,36a,36c A. Ruiz,57 J. Russ,115 V. Rusu,76

A. Safonov,119 G. Sajot,14 W.K. Sakumoto,106 Y. Sakurai,41 P. Salcido,79 A. Sánchez-Hernández,45 M. P. Sanders,26

L. Santi,38a,38b A. S. Santos,1,ii K. Sato,42 G. Savage,76 V. Saveliev,76,j A. Savoy-Navarro,76,jj L. Sawyer,89 T. Scanlon,63

R. D. Schamberger,107 Y. Scheglov,52 H. Schellman,80 P. Schlabach,76 S. Schlobohm,124 A. Schmidt,24 E. E. Schmidt,76

C. Schwanenberger,65 T. Schwarz,76 R. Schwienhorst,98 L. Scodellaro,57 A. Scribano,36a,36c F. Scuri,36a S. Seidel,103

Y. Seiya,40 J. Sekaric,87 A. Semenov,48 H. Severini,112 F. Sforza,36a,36c E. Shabalina,23 S. Z. Shalhout,69 V. Shary,18

S. Shaw,98 A.A. Shchukin,51 T. Shears,62 P. F. Shepard,116 M. Shimojima,42,nn R. K. Shivpuri,30 M. Shochet,77

I. Shreyber-Tecker,49 V. Simak,9 A. Simonenko,48 P. Sinervo,4 P. Skubic,112 P. Slattery,106 K. Sliwa,95 D. Smirnov,84

J. R. Smith,69 K. J. Smith,104 F. D. Snider,76 G. R. Snow,100 J. Snow,111 S. Snyder,108 A. Soha,76 S. Söldner-Rembold,65

H. Song,116 L. Sonnenschein,21 V. Sorin,54 K. Soustruznik,8 P. Squillacioti,36a,36c R. St. Denis,60 M. Stancari,76 J. Stark,14

B. Stelzer,4 O. Stelzer-Chilton,4 D. Stentz,76,d D.A. Stoyanova,51 M. Strauss,112 J. Strologas,103 G. L. Strycker,96

Y. Sudo,42 A. Sukhanov,76 I. Suslov,48 L. Suter,65 P. Svoisky,112 M. Takahashi,65 K. Takemasa,42 Y. Takeuchi,42 J. Tang,77

M. Tecchio,96 P. K. Teng,6 J. Thom,76,p J. Thome,115 G. A. Thompson,81 E. Thomson,114 M. Titov,18 D. Toback,119

S. Tokar,53 V.V. Tokmenin,48 K. Tollefson,98 T. Tomura,42 D. Tonelli,76 S. Torre,34 D. Torretta,76 P. Totaro,35a

M. Trovato,36a,36d Y.-T. Tsai,106 K. Tschann-Grimm,107 D. Tsybychev,107 B. Tuchming,18 C. Tully,102 F. Ukegawa,42

S. Uozumi,43 L. Uvarov,52 S. Uvarov,52 S. Uzunyan,79 R. Van Kooten,82 W.M. van Leeuwen,46 N. Varelas,78

A. Varganov,96 E.W. Varnes,67 I. A. Vasilyev,51 F. Vázquez,73,f G. Velev,76 C. Vellidis,76 P. Verdier,20 A. Y. Verkheev,48

L. S. Vertogradov,48 M. Verzocchi,76 M. Vesterinen,65 M. Vidal,85 I. Vila,57 D. Vilanova,18 R. Vilar,57 J. Vizán,57

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M. Vogel,103 P. Vokac,9 G. Volpi,34 P. Wagner,114 R. L. Wagner,76 H.D. Wahl,74 T. Wakisaka,40 R. Wallny,70

M.H. L. S. Wang,76 S.M. Wang,6 A. Warburton,4 J. Warchol,84 D. Waters,64 G. Watts,124 M. Wayne,84 J. Weichert,25

L. Welty-Rieger,80 W.C. Wester III,76 A. White,118 D. Whiteson,114,kk F. Wick,24 D. Wicke,27 A. B. Wicklund,75

E. Wicklund,76 S. Wilbur,77 H.H. Williams,114 M.R. J. Williams,61 G.W. Wilson,87 J. S. Wilson,110 P. Wilson,76

B. L. Winer,110 P. Wittich,76,p M. Wobisch,89 S. Wolbers,76 H. Wolfe,110 D. R. Wood,92 T. Wright,96 X. Wu,59 Z. Wu,122

T. R. Wyatt,65 Y. Xie,76 R. Yamada,76 K. Yamamoto,40 D. Yamato,40 S. Yang,5 T. Yang,76 U. K. Yang,77,ll W.-C. Yang,65

Y. C. Yang,43 W.-M. Yao,68 T. Yasuda,76 Y. A. Yatsunenko,48 W. Ye,107 Z. Ye,76 G. P. Yeh,76 H. Yin,76 K. Yi,76,bb K. Yip,108

J. Yoh,76 K. Yorita,41 T. Yoshida,40,mm S.W. Youn,76 G. B. Yu,109 I. Yu,43 J.M. Yu,96 S. S. Yu,76 J. C. Yun,76

A. Zanetti,38a Y. Zeng,109 J. Zennamo,104 T. Zhao,124 T.G. Zhao,65 B. Zhou,96 C. Zhou,109 J. Zhu,96

M. Zielinski,106 D. Zieminska,82 L. Zivkovic,117 and S. Zucchelli33a,33b

(CDF and D0 Collaborations)

1LAFEX, Centro Brasileiro de Pesquisas Fı́sicas, Rio de Janeiro, Brazil
2Universidade do Estado do Rio de Janeiro, Rio de Janeiro, Brazil

3Universidade Federal do ABC, Santo André, Brazil
4Institute of Particle Physics: McGill University, Montréal, Québec, Canada H3A 2T8;

Simon Fraser University, Burnaby, British Columbia, Canada V5A 1S6;
University of Toronto, Toronto, Ontario, Canada M5S 1A7;

and TRIUMF, Vancouver, British Columbia, Canada V6T 2A3
5University of Science and Technology of China, Hefei, People’s Republic of China
6Institute of Physics, Academia Sinica, Taipei, Taiwan 11529, Republic of China

7Universidad de los Andes, Bogotá, Colombia
8Charles University, Faculty of Mathematics and Physics, Center for Particle Physics, Prague, Czech Republic

9Czech Technical University in Prague, Prague, Czech Republic
10Center for Particle Physics, Institute of Physics, Academy of Sciences of the Czech Republic, Prague, Czech Republic

11Universidad San Francisco de Quito, Quito, Ecuador
12Division of High Energy Physics, Department of Physics, University of Helsinki and Helsinki Institute of Physics,

FIN-00014, Helsinki, Finland
13LPC, Université Blaise Pascal, CNRS/IN2P3, Clermont, France

14LPSC, Université Joseph Fourier Grenoble 1, CNRS/IN2P3, Institut National Polytechnique de Grenoble, Grenoble, France
15CPPM, Aix-Marseille Université, CNRS/IN2P3, Marseille, France

16LAL, Université Paris-Sud, CNRS/IN2P3, Orsay, France
17LPNHE, Universités Paris VI and VII, CNRS/IN2P3, Paris, France

18CEA, Irfu, SPP, Saclay, France
19IPHC, Université de Strasbourg, CNRS/IN2P3, Strasbourg, France

20IPNL, Université Lyon 1, CNRS/IN2P3, Villeurbanne, France;
and Université de Lyon, Lyon, France

21III. Physikalisches Institut A, RWTH Aachen University, Aachen, Germany
22Physikalisches Institut, Universität Freiburg, Freiburg, Germany

23II. Physikalisches Institut, Georg-August-Universität Göttingen, Göttingen, Germany
24Institut für Experimentelle Kernphysik, Karlsruhe Institute of Technology, D-76131 Karlsruhe, Germany

25Institut für Physik, Universität Mainz, Mainz, Germany
26Ludwig-Maximilians-Universität München, München, Germany

27Fachbereich Physik, Bergische Universität Wuppertal, Wuppertal, Germany
28University of Athens, 157 71 Athens, Greece

29Panjab University, Chandigarh, India
30Delhi University, Delhi, India

31Tata Institute of Fundamental Research, Mumbai, India
32University College Dublin, Dublin, Ireland

33aIstituto Nazionale di Fisica Nucleare Bologna, I-40127 Bologna, Italy
33bUniversity of Bologna, I-40127 Bologna, Italy

34Laboratori Nazionali di Frascati, Istituto Nazionale di Fisica Nucleare, I-00044 Frascati, Italy
35aIstituto Nazionale di Fisica Nucleare, Sezione di Padova-Trento, I-35131 Padova, Italy

35bUniversity of Padova, I-35131 Padova, Italy
36aIstituto Nazionale di Fisica Nucleare Pisa, I-56127 Pisa, Italy

36bUniversity of Pisa, I-56127 Pisa, Italy
36cUniversity of Siena, I-56127 Pisa, Italy

COMBINATION OF THE TOP-QUARK MASS . . . PHYSICAL REVIEW D 86, 092003 (2012)

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36dScuola Normale Superiore, I-56127 Pisa, Italy
37aIstituto Nazionale di Fisica Nucleare, Sezione di Roma 1, I-00185 Roma, Italy

37bSapienza Università di Roma, I-00185 Roma, Italy
38aIstituto Nazionale di Fisica Nucleare Trieste/Udine, I-34100 Trieste, Italy

38bUniversity of Udine, I-33100 Udine, Italy
39Okayama University, Okayama 700-8530, Japan

40Osaka City University, Osaka 588, Japan
41Waseda University, Tokyo 169, Japan

42University of Tsukuba, Tsukuba, Ibaraki 305, Japan
43Center for High Energy Physics: Kyungpook National University, Daegu 702-701, Korea;

Seoul National University, Seoul 151-742, Korea;

Sungkyunkwan University, Suwon 440-746, Korea;

Korea Institute of Science and Technology Information, Daejeon 305-806, Korea;

Chonnam National University, Gwangju 500-757, Korea; Chonbuk National University, Jeonju 561-756, Korea
44Korea Detector Laboratory, Korea University, Seoul, Korea

45CINVESTAV, Mexico City, Mexico
46Nikhef, Science Park, Amsterdam, Netherlands

47Radboud University Nijmegen, Nijmegen, Netherlands
48Joint Institute for Nuclear Research, Dubna, Russia

49Institute for Theoretical and Experimental Physics, Moscow, Russia
50Moscow State University, Moscow, Russia

51Institute for High Energy Physics, Protvino, Russia
52Petersburg Nuclear Physics Institute, St. Petersburg, Russia

53Comenius University, 842 48 Bratislava, Slovakia; and Institute of Experimental Physics, 040 01 Kosice, Slovakia
54Institut de Fisica d’Altes Energies, ICREA, Universitat Autonoma de Barcelona, E-08193, Bellaterra (Barcelona), Spain

55Institució Catalana de Recerca i Estudis Avançats (ICREA) and Institut de Fı́sica d’Altes Energies (IFAE), Barcelona, Spain
56Centro de Investigaciones Energeticas Medioambientales y Tecnologicas, E-28040 Madrid, Spain

57Instituto de Fisica de Cantabria, CSIC-University of Cantabria, 39005 Santander, Spain
58Uppsala University, Uppsala, Sweden

59University of Geneva, CH-1211 Geneva 4, Switzerland
60Glasgow University, Glasgow G12 8QQ, United Kingdom
61Lancaster University, Lancaster LA1 4YB, United Kingdom

62University of Liverpool, Liverpool L69 7ZE, United Kingdom
63Imperial College London, London SW7 2AZ, United Kingdom

64University College London, London WC1E 6BT, United Kingdom
65The University of Manchester, Manchester M13 9PL, United Kingdom

66University of Oxford, Oxford OX1 3RH, United Kingdom
67University of Arizona, Tucson, Arizona 85721, USA

68Ernest Orlando Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA
69University of California, Davis, Davis, California 95616, USA

70University of California, Los Angeles, Los Angeles, California 90024, USA
71University of California, Riverside, Riverside, California 92521, USA

72Yale University, New Haven, Connecticut 06520, USA
73University of Florida, Gainesville, Florida 32611, USA

74Florida State University, Tallahassee, Florida 32306, USA
75Argonne National Laboratory, Argonne, Illinois 60439, USA

76Fermi National Accelerator Laboratory, Batavia, Illinois 60510, USA
77Enrico Fermi Institute, University of Chicago, Chicago, Illinois 60637, USA

78University of Illinois at Chicago, Chicago, Illinois 60607, USA
79Northern Illinois University, DeKalb, Illinois 60115, USA
80Northwestern University, Evanston, Illinois 60208, USA

81University of Illinois, Urbana, Illinois 61801, USA
82Indiana University, Bloomington, Indiana 47405, USA

83Purdue University Calumet, Hammond, Indiana 46323, USA
84University of Notre Dame, Notre Dame, Indiana 46556, USA

85Purdue University, West Lafayette, Indiana 47907, USA
86Iowa State University, Ames, Iowa 50011, USA

87University of Kansas, Lawrence, Kansas 66045, USA
88Kansas State University, Manhattan, Kansas 66506, USA
89Louisiana Tech University, Ruston, Louisiana 71272, USA

90The Johns Hopkins University, Baltimore, Maryland 21218, USA

T. AALTONEN et al. PHYSICAL REVIEW D 86, 092003 (2012)

092003-4



91Boston University, Boston, Massachusetts 02215, USA
92Northeastern University, Boston, Massachusetts 02115, USA
93Harvard University, Cambridge, Massachusetts 02138, USA

94Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
95Tufts University, Medford, Massachusetts 02155, USA

96University of Michigan, Ann Arbor, Michigan 48109, USA
97Wayne State University, Detroit, Michigan 48201, USA

98Michigan State University, East Lansing, Michigan 48824, USA
99University of Mississippi, University, Mississippi 38677, USA

100University of Nebraska, Lincoln, Nebraska 68588, USA
101Rutgers University, Piscataway, New Jersey 08855, USA
102Princeton University, Princeton, New Jersey 08544, USA

103University of New Mexico, Albuquerque, New Mexico 87131, USA
104State University of New York, Buffalo, New York 14260, USA
105The Rockefeller University, New York, New York 10065, USA
106University of Rochester, Rochester, New York 14627, USA

107State University of New York, Stony Brook, New York 11794, USA
108Brookhaven National Laboratory, Upton, New York 11973, USA

109Duke University, Durham, North Carolina 27708, USA
110The Ohio State University, Columbus, Ohio 43210, USA
111Langston University, Langston, Oklahoma 73050, USA

aDeceased.
bD0 visitors from Augustana College, Sioux Falls, SD, USA.
cWith CDF visitors from Universidad de Oviedo, E-33007 Oviedo, Spain.
dWith CDF visitors from Northwestern University, Evanston, IL 60208, USA.
eD0 visitors from The University of Liverpool, Liverpool, United Kingdom.
fWith CDF visitors from Universidad Iberoamericana, Mexico D.F., Mexico.
gWith CDF visitors from ETH, 8092 Zurich, Switzerland.
hWith CDF visitors from CERN, CH-1211 Geneva, Switzerland.
iWith CDF visitors from Queen Mary, University of London, London, E1 4NS, United Kingdom.
jWith CDF visitors from National Research Nuclear University, Moscow, Russia.
kWith CDF visitors from Yarmouk University, Irbid 211-63, Jordan.
lWith CDF visitors from Muons, Inc., Batavia, IL 60510, USA.

mD0 visitors from UPIITA-IPN, Mexico City, Mexico.
nD0 visitors from DESY, Hamburg, Germany.
oD0 visitors from University College London, London, United Kingdom.
pWith CDF visitors from Cornell University, Ithaca, NY 14853, USA.
qWith CDF visitors from Kansas State University, Manhattan, KS 66506, USA.
rWith CDF visitors from Kinki University, Higashi-Osaka City, Japan 577-8502.
sWith CDF visitors from University of California Santa Barbara, Santa Barbara, CA 93106, USA.
tWith CDF visitors from University of Notre Dame, Notre Dame, IN 46556, USA.
uWith CDF visitors from Ewha Womans University, Seoul, 120-750, Korea.
vWith CDF visitors from Texas Tech University, Lubbock, TX 79609, USA.
wWith CDF visitors from University of Melbourne, Victoria 3010, Australia.
xD0 visitors from Centro de Investigacion en Computacion-IPN, Mexico City, Mexico.
yWith CDF visitors from Institute of Physics, Academy of Sciences of the Czech Republic, Czech Republic.
zWith CDF visitors from Istituto Nazionale di Fisica Nucleare, Sezione di Cagliari, 09042 Monserrato (Cagliari), Italy.
aaWith CDF visitors from University College Dublin, Dublin 4, Ireland.
bbWith CDF visitors from University of Iowa, Iowa City, IA 52242, USA.
ccWith CDF visitors from University of California Santa Cruz, Santa Cruz, CA 95064, USA.
ddD0 visitors from SLAC, Menlo Park, CA, USA.
eeD0 visitors from ECFM, Universidad Autonoma de Sinaloa, Culiacán, Mexico.
ffWith CDF visitors from Universidad Tecnica Federico Santa Maria, 110v Valparaiso, Chile.
ggWith CDF visitors from University of Cyprus, Nicosia CY-1678, Cyprus.
hhWith CDF visitors from Office of Science, U.S. Department of Energy, Washington, DC 20585, USA.
iiD0 visitors from Universidade Estadual Paulista, São Paulo, Brazil.
jjWith CDF visitors from CNRS-IN2P3, Paris, F-75205 France.
kkWith CDF visitors from University of California Irvine, Irvine, CA 92697, USA.
llWith CDF visitors from University of Manchester, Manchester M13 9PL, United Kingdom.

mmWith CDF visitors from University of Fukui, Fukui City, Fukui Prefecture, Japan 910-0017.
nnWith CDF visitors from Nagasaki Institute of Applied Science, Nagasaki, Japan.
ooWith CDF visitors from Nagasaki Institute of Applied Science, Nagasaki, Japan.

COMBINATION OF THE TOP-QUARK MASS . . . PHYSICAL REVIEW D 86, 092003 (2012)

092003-5



112University of Oklahoma, Norman, Oklahoma 73019, USA
113Oklahoma State University, Stillwater, Oklahoma 74078, USA

114University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA
115Carnegie Mellon University, Pittsburgh, Pennsylvania 15213, USA

116University of Pittsburgh, Pittsburgh, Pennsylvania 15260, USA
117Brown University, Providence, Rhode Island 02912, USA

118University of Texas, Arlington, Texas 76019, USA
119Texas A&M University, College Station, Texas 77843, USA
120Southern Methodist University, Dallas, Texas 75275, USA

121Rice University, Houston, Texas 77005, USA
122Baylor University, Waco, Texas 76798, USA

123University of Virginia, Charlottesville, Virginia 22904, USA
124University of Washington, Seattle, Washington 98195, USA
125University of Wisconsin, Madison, Wisconsin 53706, USA

(Received 4 July 2012; published 2 November 2012)

The top quark is the heaviest known elementary particle, with a mass about 40 times larger than the

mass of its isospin partner, the bottom quark. It decays almost 100% of the time to aW boson and a bottom

quark. Using top-antitop pairs at the Tevatron proton-antiproton collider, the CDF and D0 Collaborations

have measured the top quark’s mass in different final states for integrated luminosities of up to 5:8 fb�1.

This paper reports on a combination of these measurements that results in a more precise value of the mass

than any individual decay channel can provide. It describes the treatment of the systematic uncertainties

and their correlations. The mass value determined is 173:18� 0:56 ðstatÞ � 0:75 ðsystÞ GeV or

173:18� 0:94 GeV, which has a precision of �0:54%, making this the most precise determination of

the top-quark mass.

DOI: 10.1103/PhysRevD.86.092003 PACS numbers: 14.65.Ha, 13.85.Ni, 13.85.Qk, 12.15.Ff

I. INTRODUCTION

A. The top quark

The standard model (SM) of particle physics describes
the elementary particles and their interactions. The top
quark (t) has a special place in the hierarchy of particles
because it is far more massive than any of the other
fundamental objects. It is the up-type quark, partnered
with the down-type bottom quark (b), forming the third
generation of quarks that was predicted by Kobayashi and
Maskawa in 1973 [1] to accommodate CP violation in
neutral kaon decays [2]. At particle colliders the top quark
is produced mainly in top-antitop (t�t) pairs. The first evi-
dence of top-quark production was reported by the CDF
Collaboration [3], and the top quark was first observed
in this production mode by the CDF [4] and D0 [5]
Collaborations at the Tevatron proton-antiproton collider.
Since then, great efforts have been focused on measuring
its properties with ever higher precision. In addition to its
large mass (mt), the top quark is also singular because it
decays before it can hadronize: there are no mesons or
baryons containing valence top quarks. The top quark
decays almost exclusively to a W boson and a b quark,
with the fraction determined by the near-unity value of the
Cabibbo-Kobayashi-Maskawa (CKM) quark mixing ma-
trix [1,6] element Vtbð� 0:9992Þ [7]. Its other decays are
limited by the small values of Vts � 0:0387 and Vtd �
0:0084 [7], assuming three-family unitarity of the CKM
matrix. The W boson decays to a charged lepton and its

associated neutrino, or to a quark-antiquark pair, and the
final states of t�t events are thus characterized as follows:
‘‘leptonþ jets’’ (t�t!‘þ�bq �q0 �b and �qq0b‘� �� �b); ‘‘alljets’’
(t�t ! q �q0b �qq0 �b), and ‘‘dileptons’’ (t�t ! ‘þ�b‘� �� �b ). In
this notation the charged lepton ‘ represents an electron or
muon, and q is a first- or second-generation quark. The W
boson also decays to a � lepton and a � neutrino. If �
decays to an electron or muon, the event contributes to the
lepton categories, and if the � decays into hadrons, it
contributes to the leptonþ jets or alljets categories. A
fourth category labeled ‘‘ 6ET þ jets’’ is used to measure
mt when there are jets and a large imbalance in transverse
momentum in the event ( 6ET), but no identified lepton. It
comprises t�t ! �þ�b�� �� �b , �þ�bq �q0 �b, and �qq0b�� �� �b
final states, accounting for 40% of the t�t signal events in
the 6ET þ jets category, or ‘þ�bq �q0 �b, �qq0b‘� �� �b , where
the electron or muon are not reconstructed, accounting for
60% of the t�t signal in this category. Additional contribu-
tions to 6ET arise from the neutrino(s) produced in � decays.
In dilepton events, there are typically two jets from the

two b quarks, one from each top-quark decay. In leptonþ
jets events, there are typically four jets, including two b jets
and two light-quark jets from W-boson decay. Alljets
events most often contain six jets, the two b jets and four
light-quark jets. The 6ET þ jets events usually have four or
five jets. Additional gluon or quark jets can arise owing to
radiation from initial or final-state colored particles, in-
cluding the top quarks. About 23% of the t�t events have an
extra jet with sufficient energy to pass the selection criteria,

T. AALTONEN et al. PHYSICAL REVIEW D 86, 092003 (2012)

092003-6

http://dx.doi.org/10.1103/PhysRevD.86.092003


and about 5% of the events have two additional jets. These
extra jets complicate the measurement of mt and degrade
its resolution. Figure 1 illustrates leading-order (LO) pro-
duction of t�t events at the Fermilab Tevatron Collider, and
Fig. 2 shows the relevant t�t decay modes.

B. Top-quark mass origin and definitions

One of the fundamental properties of an elementary
particle is its mass. In the SM, fermions acquire mass
through interactions with the Higgs field [8]. Absolute
values of these masses are not predicted by the SM. In
theoretical calculations, a particle’s mass can be defined in
more than one way, and it depends on how higher-order
terms in perturbative quantum chromodynamics (QCD)
calculations are renormalized. In the modified minimal

subtraction scheme (MS), for example, the mass definition
reflects short-distance effects, whereas in the pole-mass
scheme the mass definition reflects long-distance effects
[9]. The concept of the pole mass is not well defined since
color confinement does not provide S-matrix poles at
m ¼ mt [10]. Direct mass measurements that are inputs
to the combination described in this paper rely on
Monte Carlo (MC) generators to extract mt. Hence the
measured mass corresponds in fact to the mass parameter
in the MC. Work is proceeding to address the exact differ-

ence between the measured mass and the pole mass, as
presented, for example, in Appendix C of Ref. [11]. One
alternative way to address this problem is to extract mt

from a measurement of the t�t cross section [12]. The D0
Collaboration has recently shown that the directly mea-
sured mass of the top quark is closer to the pole mass
extracted from a measurement of the t�t cross section than

to an MS mass extracted in a similar way [12]. Hence,
within the precision of theory and data, the directly mea-
sured mt is best interpreted as the top-quark pole mass.
CPT invariance predicts that a particle and its antipar-

ticle partner have the same mass. This has been checked for
the top quark by the D0, CDF, and CMS Collaborations,
and the masses are found to hold within the measurement
uncertainties, with �mt ¼ mt �m�t ¼ 0:84� 1:87 GeV
[13], �mt ¼ �3:3� 1:7 GeV [14], and �mt ¼ �0:44�
0:53 GeV [15], respectively. Thus, the top-quark mass
combination in this paper assumes mt ¼ m�t.

C. Predictions based on the top-quark mass

The internal consistency of the SM can be tested by
using different observables to predict the values of others
and then to compare the expectations with their measured
values. For example, the relation between the mass of the
W boson (MW) and sin

2�W (the electroweak mixing angle)
includes higher-order radiative corrections involving mt;
hence the smaller the uncertainty on the measured mt, the
stronger is the test of consistency.
Since 1997, the LEP Electroweak Working Group has

used the observed top-quark and the W boson masses and
other precision electroweak variables to extract constraints
on the Higgs boson mass (MH) in the SM [16]. This has
been extended to the minimal supersymmetric standard
model [17], and the GFITTER Collaboration has applied
the technique to set limits on a wide variety of theories
beyond the SM [18]. Figure 3(a) shows the combined
constraint attributable to MW and mt (as of March 2012)
on the Higgs boson mass. Figure 3(b) shows the constraint
from MW and mt separately (as of March 2012) on the
Higgs boson mass, and a global constraint originating from
all the other electroweak variables, showing the impor-
tance of the MW and mt variables to constrain the Higgs
boson mass.

D. History of measurement of mt

Before 1995, global fits to electroweak data from the
CERN and SLAC eþe� colliders (LEP and SLC) and from
other experiments produced estimates of mt that ranged
from� 90 GeV to� 190 GeV [19]. At the time of the first
observation of the top quark in 1995, the fits indicated a
mass close to the current Tevatron value of mt, but with an
uncertainty of � �10% and an assumption of 300 GeV
mass of the Higgs boson [20]. CDF measured mt ¼ 176�
8 ðstatÞ � 10 ðsystÞ GeV [4] (total uncertainty of 7%) and

g

g

t

t

g

g

t

t

q

q

t

t

FIG. 1 (color online). Examples of tree Feynman diagrams for
t�t production. At the Tevatron collider, the q �q channel contrib-
utes 81% to the total t�t inclusive cross section and the gg channel
the remaining 19% [69,96].

t

t

g

b

b

W +

W

+ + +

FIG. 2 (color online). Leading-order Feynman diagram for t�t
decay. The dilepton modes (ee, e�, ��) have a combined
branching fraction of � 4%, the electronþ jets and muonþ
jets modes combined correspond to� 30%, and the alljets mode
has a branching fraction of � 46%. The � modes are shared
among the 6ET þ jets and the other channels in the analyses.

COMBINATION OF THE TOP-QUARK MASS . . . PHYSICAL REVIEW D 86, 092003 (2012)

092003-7



D0 measured mt¼199þ19
�21ðstatÞ�22ðsystÞGeV [5] (total

uncertainty of 15%).
Since then, the CDF and D0 Collaborations have devel-

oped many novel measurement techniques and published
nearly 50 journal papers on their measurements of mt.
Recently, the CMS Collaboration at the Large Hadron
Collider (LHC) published a measurement using 102
dilepton events [21] and finds mt¼175:5�4:6ðstatÞ�
4:6ðsystÞGeV (total uncertainty of 3.7%). The ATLAS
Collaboration at the LHC has submitted a measurement
of mt ¼ 174:5� 0:6� 2:3 GeV (total uncertainty of
1.4%) using nearly 12,000 leptonþ jets events [22]. The
most precise measurements from the Tevatron in a single
decay channel use leptonþ jets events, a matrix-element

method as introduced in Ref. [23], and an in situ calibration
of the jet energy scale. CDF’s matrix-element measure-
ment [24] uses 5:6 fb�1 of integrated luminosity to find
mt ¼ 173:00� 0:65 ðstatÞ � 1:06 ðsystÞ GeV (total uncer-
tainty of 0.72%). D0’s measurement [25] uses 3:6 fb�1 of
integrated luminosity to obtain mt¼174:94�0:83ðstatÞ�
1:24ðsystÞGeV (total uncertainty of 0.85%). Figure 4
shows the publication history of the direct measurements
of mt at the Tevatron.

E. Overview of mass measurements

This paper reports on the combination of previously
published measurements of mt. Details of the analyses
are therefore not repeated as this information is available
in recent reviews [26], as well as in the publications of each
of the results. We will, however, summarize the basic
techniques used for the measurements.
The cross section for t�t production in proton-antiproton

(p �p) interactions at 1.96 TeV is � 7:2 pb [27,28]. The
mean transverse momentum (pT) of the t�t system at parton
level is � 20 GeV, which is attributed to initial-state
radiation (i.e., gluon emission). The mean transverse mo-
mentum of the top quarks at parton level is� 95 GeV [29].
Top quarks have a lifetime of � 0:3� 10�24 s [30,31],
which is an order of magnitude smaller than the time scale
for parton evolution and hadronization. Hence, when top
quarks decay, they transfer their kinematic characteristics
to the W boson and b quark, and the measured energy-
momentum four-vectors of the final-state particles can be
used to reconstruct the mass of the top quark, except for the
presence of initial or final-state radiation.
In alljets events, the four-vector of every jet emerging

from quarks can be reconstructed, but neutrinos emitted in
semileptonic decays of b quarks and jet energy resolution
effects will lead to lost energy. In leptonþ jets events, the
momentum of the neutrino from theW ! ‘�‘ decay is not
detected. The transverse component can be inferred from
the negative of the vector sum of all transverse momenta of
particles detected in the calorimeter and muon detectors.
We estimate the longitudinalmomentumof�‘ by constrain-
ing the mass of the charged lepton and neutrino system to
the world average value of MW [7]. We also use MW to
choose the two light jets fromW ! q �q0 decay, and we use
that information for an in situ calibration of jet energies.
In dilepton events, the analysis is more complicated
because there are two final-state neutrinos from the leptonic
decays of both W bosons. Therefore, the longitudinal and
transverse-momentum components of the neutrinos cannot
be determinedwithout the application ofmore sophisticated
tools. These involve assuming a value for mt to solve the
event kinematics and assigning a weight to each mt hy-
pothesis to determine the most likely value ofmt consistent
with the hypothesis that the event is a t�t event.
A major issue in t�t final-state reconstruction is the

correct mapping of the reconstructed objects to the partons

Not excluded at 95% C.L. by direct searches

: Tevatron
t

: LEP+Tevatron, mWM

 <
 1

27
 G

eV

H

11
5.

5 
< 

M

 >
 6

00
 G

eV

HM

68% C.L.

 [GeV]tm

 [
G

eV
]

W
M

80.30

80.32

80.34

80.36

80.38

80.40

80.42

80.44

80.46

(a)

165 170 175 180 185 190

 [GeV]tm
140 150 160 170 180 190 200

 [
G

eV
]

W
M

80.2

80.3

80.4

80.5

15
=127 GeV

W band for LEP+Tevatron M1

=1
.5 GeV

HM HM =600 GeV

HM =1000 GeV

HM

G fitter SM

M
ay 12

top

 band for1
Tevatron m

68%, 95%, 99% CL fit

top
, mWcontours excl. M

(b)

FIG. 3 (color online). (a) Constraints from LEP and Tevatron
measurements of MW and mt (Tevatron only) on MH within the
SM. The regions in the mass of the Higgs boson still allowed
after the direct searches at LEP, Tevatron, and LHC are also
shown. (b) From Ref. [18], the large countors (blue) indicate the
constraints on the Higgs boson, from global fits to electroweak
data without including the direct measurements of MW and mt

from the Tevatron.

T. AALTONEN et al. PHYSICAL REVIEW D 86, 092003 (2012)

092003-8



from the decays of the top quark and W boson. The prob-
lem arises because often the jet charge and flavor cannot be
uniquely determined. This creates combinatorial ambigu-
ities in the t�t event reconstruction that vary from 90
possible jet-to-parton assignments for the alljets final state
to 2 in the dilepton channel. In the leptonþ jets and
dilepton final states, additional ambiguities may arise
from multiple kinematical solutions for the longitudinal
component of the neutrino momentum.

Two methods are used to measure the value ofmt. In the
firstmethod, the reconstructedmass distribution in data, or a
variable correlated withmt, such as the decay length of the
B hadron or the transverse momentum of a lepton, is com-
pared to template distributions composed of contributions
from background and simulation of t�t events. One template
is used to represent background and another for each puta-
tivevalue ofmt. The secondmethod uses event probabilities
based on the LOmatrix element for the production of t�t. For
each event, a probability is calculated as a function of mt

that this event is from t�t production, as based on the corre-
sponding production and decay matrix element. Detector
resolution is taken into account in the calculation of these
probabilities through transfer functions that correlate
parton-level energies and their measured values. The value
ofmt is then extracted from the joint probability calculated
for all selected events, based on the probability for signal
and background (also defined through its matrix element).
This method produces the most accurate results, but the
computations are time consuming.

F. Combination overview

This paper describes the combination of statistically
independent top-quark mass measurements from the

Fermilab TevatronCollider.Measurements are independent
if they are based on different data sets, e.g., from CDF and
from D0, or from Tevatron Run I (1992–1996) and Run II
(2001–2011). They are also independent within one data set
if the event selections are designed to be exclusive; i.e., no
event can pass more than one category of selections. At
times, more than one measurement is published using the
same data and decay channel. In this situation, the result
with smallest overall uncertainty is chosen for the combi-
nation. Twelve measurements are used in the combination
described here, eight from the CDF collaboration and four
from D0. These comprise five leptonþ jets measurements
(CDF and D0, Run II and Run I, and a CDF Run II result
based on the decay length of B hadrons); two alljets mea-
surements (CDF Run II and Run I); four dilepton measure-
ments (CDF and D0, Run II and Run I); and a 6ETþjets
measurement (CDF Run II). We combine these measure-
ments using an analytic method called the best linear
unbiased estimator (BLUE) [32–34]. This technique forms
a linear combination of the separate unbiased mass mea-
surements to produce the best estimate ofmt with the small-
est uncertainty. This procedure follows a series of 11 such
mass combinations presented in [35–45], updated each year
since 2004 as new measurements of mt became available.
The combination presented here is the first to be published
in a peer-reviewed journal.

II. INPUTS TO THE COMBINATION

A. The independent mass measurements

The mass measurements included in the combination are
shown in Table I [24,25,46–55]. These 12 channels are
chosen because they are statistically independent, which

210

205

200

195

190

185

180

175

170

165

160

155

150

19
93

19
94

19
95

19
96

19
97

19
98

19
99

20
00

20
01

20
02

20
03

20
04

20
05

20
06

20
07

20
08

20
09

20
10

20
11

20
12

Publication Date

T
o

p
 Q

u
ar

k 
M

as
s 

  [
G

eV
] CDF

D   

Lepton+jets
Alljets

Dileptons
ET +jets

Decay length
Lepton+track

Lepton+jets + dilepton
Tevatron combination

Tevatron Run I Tevatron Run II

Publications from Run I Publications from Run II

FIG. 4 (color online). The CDF and D0 published direct measurements of the top-quark mass as a function of time.

COMBINATION OF THE TOP-QUARK MASS . . . PHYSICAL REVIEW D 86, 092003 (2012)

092003-9



maximizes the improvement in the combination, and be-
cause enough information is available to separate the
components of systematic uncertainty for proper treatment
in the combination.

The D0 measurement from 2005 in the alljets channel
(Run I) [56] of mt ¼ 178:5� 13:7 ðstatÞ � 7:7 ðsystÞ GeV
(total uncertainty of 8.8%) is not included in the combina-
tion because some subcomponents of the systematic
uncertainty are not available.

The CDF measurement from Run II based on decay-
length analysis [55] differs from the others in that it uses
the mean decay length of B hadrons in b-tagged leptonþ
jets events as the mt-sensitive variable. It is independent of
energy information in the calorimeter, and its main source
of systematic uncertainty is uncorrelated with the dominant
ones from the jet energy scale calibration in other
measurements. This measurement of mt is essentially
uncorrelated with the higher precision CDF result from
the leptonþ jets channel. The overlap between the data
samples used for the decay-length method and the
leptonþ jets sample has therefore no effect.

B. Data

The data were collected with the CDF [57] and D0
[58,59] detectors at the Tevatron p �p collider at Fermilab
between 1992 and 2009. The Tevatron ‘‘center-of-mass’’
energy was 1.8 TeV in Run I from 1992 to 1996 and
1.96 TeV in Run II from 2001. A silicon microstrip tracker
around the beam pipe at the center of each detector was
used to reconstruct charged-particle tracks (only in Run II
at D0). Tracks spatially matched to calorimeter jets are
checked for originating from a secondary vertex, or for
evidence that they originate from decays of long-lived
heavy-flavor hadrons containing b quarks from the decay
of top quarks [57,60]. Electrons and jets produce particle

showers in the calorimeters, and the collected information
is used to measure their energies. Muons traverse the
calorimeters and outer muon detectors that are used to
reconstruct their tracks. Both CDF and D0 have central
axial magnetic fields in the tracking region (D0 only in
Run II), in which the momenta of charged particles are
determined from the curvature of their tracks. The CDF
magnet has a diameter of 3 m and extends 4.8 m along the
beam line, with a field strength of 1.4 T, and the D0 magnet
has a diameter of 1.0 m and length of 2.7 m to fit inside the
Run I calorimeter with a field strength of 2.0 T. The CDF
detector’s larger tracking volume with a higher density of
measurements gives better transverse-momentum resolu-
tion for charged-particle tracks. The transverse-momentum
resolution is� 3:5% at CDF and� 10% at D0 for a muon
with pT ¼ 50 GeV. The trigger and event-selection
criteria depend on the t�t final states, details of which appear
in the publications listed in Table I. The experiments
collected Oð1014Þ hard collisions, from which 7420 events
are selected because they have the characteristics expected
for t�t pairs, of which � 56% are expected to be true t�t
events.

C. Models for t �t signal

The t�t signal in Run I was simulated using the LO
generator HERWIG [61] with the MRSD0

0 [62] and

CTEQ4M [63] parton distribution functions (PDF) used
by CDF and D0, respectively. The HERWIG generator im-
plements the hard-scattering processes q �q ! t�t and
gg ! t�t, adding initial-state and final-state radiation
through leading-log QCD evolution [64]. The top quark
and W boson in HERWIG decay according to the branching
fractions listed by the Particle Data Group [7], and the
final-state partons are subsequently fragmented into
jets. The MC events are then processed through a fast

TABLE I. Top-quark mass measurements used as input to determine the combined value of mt from the Tevatron and the combined
result.

Decay channel

or method

Tevatron

period Experiment

Integrated

luminosity

[fb�1]

Number

of events

Background

[%] mt [GeV]

Uncertainty

on mt [%] Reference

Leptonþ jets Run II CDF 5.6 1087 17 173:00� 0:65� 1:06 0.72 [24]

Leptonþ jets Run II D0 3.6 615 27 174:94� 0:83� 1:24 0.85 [25]

Leptonþ jets Run I CDF 0.1 76 54 176:1� 5:1� 5:3 4.2 [46]

Leptonþ jets Run I D0 0.1 22 22 180:1� 3:6� 3:9 2.9 [47]

Alljets Run II CDF 5.8 2856 71 172:47� 1:43� 1:40 1.2 [48]

Alljets Run I CDF 0.1 136 79 186:0� 10:0� 5:7 6.2 [49]

Dileptons Run II CDF 5.6 392 23 170:28� 1:95� 3:13 2.2 [50]

Dileptons Run II D0 5.3 415 21 174:00� 2:36� 1:44 1.6 [51]

Dileptons Run I CDF 0.1 8 16 167:4� 10:3� 4:9 6.8 [52]

Dileptons Run I D0 0.1 6 25 168:4� 12:3� 3:6 7.6 [53]

6ET þ jets Run II CDF 5.7 1432 32 172:32� 1:80� 1:82 1.5 [54]

Decay length Run II CDF 1.9 375 30 166:90� 9:00� 2:82 5.7 [55]

Combination � 5:8 7420 44 173:18� 0:56� 0:75 0.54

T. AALTONEN et al. PHYSICAL REVIEW D 86, 092003 (2012)

092003-10



simulation or a GEANT model [65] of the detectors and then
through event reconstruction programs.

For the t�t signal in Run II, CDF uses PYTHIA [66] with
the CTEQ5L [67] PDF, and D0 uses the leading-log gen-
erator ALPGEN [68] with the CTEQ6L1 [69] PDF and
PYTHIA for parton showering. ALPGEN contains more

tree-level graphs in higher-order �s than PYTHIA. ALPGEN
has parton-jet matching [70], which avoids double count-
ing of partons in overlapping regions of jet kinematics.
CDF sets the event generation factorization and renormal-
ization scales Q2 to m2

t þ p2
? þ ðP2

1 þ P2
2Þ=2, where p? is

the transverse momentum characterizing the scattering
process, and P2

1 and P2
2 are the virtualities of the incoming

partons. D0 sets the scales to m2
t þ hp2

Ti, where hp2
Ti is the

average of the square of transverse momentum of all other
light partons produced in association with the t�t pair. The
PYTHIA model treats each step of the t�t decay chain

(t ! Wb, W ! ‘� or q �q0) separately and does not pre-
serve spin correlations. ALPGEN uses exact matrix elements
for each step and thereby correctly describes the spin
information of the final-state partons. The fragments of
the proton and antiproton or ‘‘underlying event’’ are added
separately to each hard collision. CDF uses the ‘‘Tune A’’
settings [71] in PYTHIA while D0 uses a modified version
of the tune. Both collaborations use angular ordering
for modeling parton showering in PYTHIA, and not
pT-ordered models. The underlying event is therefore
not interleaved with the parton showers as in models of
color reconnection [72].

D. Background models

In the leptonþ jets channel, the dominant background is
from W þ jets production. Smaller contributions arise
from multijet events, Zþ jets, single top-quark (tqb and
tb), and diboson production (WW, WZ, and ZZ). The
alljets channel has mainly multijet events as background.
The largest background in the dilepton channel is from
Zþ jets events, which include Drell-Yan production.
Backgrounds from diboson production and from events
with jets identified as leptons are very small in the dilepton
channel. The 6ET þ jets channel has multijet events and
W þ jets as main backgrounds.

In all channels contributions from multijet events are
modeled using data. Most other background sources are
modeled through MC simulation. In Run I, both collabo-
rations used VECBOS [73] to model W þ jets events.
VECBOS is a precursor of ALPGEN and provides one

of the first models of events with many high-momentum
final-state partons. PYTHIA was used to model Zþ jets,
Drell-Yan, and diboson processes. Background from
events with a single top quark was negligible. In Run II,
both collaborations used ALPGEN for the simulation of the
W þ jets background. The treatment of heavy-flavor jets is
implemented more accurately in ALPGEN, and parton-jet
matching also improves the simulation. For the Zþ jets

background, CDF uses PYTHIA and D0 uses ALPGEN. For
dibosons, both collaborations use PYTHIA. Processes with a
single top quark are modeled by CDF using MADEVENT

[74] (based on MADGRAPH [75]) and by D0 with SINGLETOP

[76] (based on COMPHEP [77]).
The uncertainty in the description of the W þ jets back-

ground has three main components: (i) the uncertainty on
the scale Q2, which affects both the overall normalization
and the differential jet distributions in pseudorapidity �
[78] and pT ; (ii) the uncertainty in the correction for flavor
content of jets to higher order; and (iii) the limitation in the
MC model we are using to reproduce the jet pT and �
distributions in data at low pT and large j�j.

E. Jet properties

After the top quarks decay, the final-state quarks and
gluons hadronize to produce multiple charged and neutral
particles that traverse the central tracking systems into the
calorimeters, where they produce many lower-momentum
particles through interactions in the absorbers of the calo-
rimeters. The observed particles tend to cluster in jets that
can be assigned to the initial partons. For jet reconstruc-
tion, the CDF Collaboration uses a clustering algorithm in
ð�;�Þ space [79] with a cone radius of

CDF R ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ð��Þ2 þ ð��Þ2

q
¼ 0:4;

where � is the azimuthal angle around the beam line, � is
the pseudorapidity, and �� or �� are the widths of the
cone. D0 uses a midpoint iterative seed-based cone algo-
rithm in ðy;�Þ space [80] with a radius defined by

D0 R ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ð�yÞ2 þ ð��Þ2

q
¼ 0:5;

where the rapidity y ¼ 1=2 lnððEþ pLÞ=ðE� pLÞÞ, E is
the jet energy, and pL is its longitudinal momentum
component.
The jet energy resolution in the central region (j�j< 1)

is approximately the same for CDF and D0; for CDF it is

�ðETÞ=ET ¼ 50%=
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ET ðGeVÞp � 3%. For jets in the

forward region, however, the energy resolution at D0 is
similar to that in the central region, while at CDF it is not as

good [�ðETÞ=ET ¼ 70%=
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ET ðGeVÞp � 4%]. CDF’s calo-

rimeter covers j�j< 3:8, whereas D0’s calorimeter covers
j�j< 4:2. The D0 calorimeter is more homogeneous, so
that the imbalance in transverse momentum (see Sec. II G)
usually has better resolution at D0. For both CDF and D0,
to reject jets with mismeasured energy, selections on en-
ergy deposition are required when clustering the energy
from the calorimeter cells into jets. When a muon is
reconstructed within the jet cone, a correction is applied
to the jet energy to account for the muon and its associated
neutrino assumed to arise from heavy-quark decay.
Jet energy scale calibrations are applied after jet recon-

struction. CDF calibrates the transverse momentum using

COMBINATION OF THE TOP-QUARK MASS . . . PHYSICAL REVIEW D 86, 092003 (2012)

092003-11



test-beam data and single-particle simulated events and
corrects the jet energy to the parton level. Consequently,
CDF does not calibrate the jet energy scale in MC events.
D0 calibrates the energy using photonþ jets and two-jet
data and calibrates jets in data as well as in MC to the
observed particle level. Particle jets are clustered from
stable particles after fragmentation, including particles
from the underlying event, but excluding undetected
energy from muons and neutrinos.

CDF’s jet calibration [81] applies two scale factors and
three offsets to convert the measured transverse momen-
tum of a jet to that of the parton that initiated the jet. D0’s
jet calibration [82] applies three scale factors and one
offset to the jet energy to convert to the particle jet energy
scale. The calibrations are expressed as follows:

CDF pparton
T ¼ p

jet
T Rrel � CMI

Rabs

� CUE þ COC;

D0 Eparticle ¼ Ejet � CMI;UE

RabsRrelFOC

:

The absolute response Rabs corrects for energy lost in
uninstrumented regions between calorimeter modules, for
differences between electromagnetically and hadronically
interacting particles, as well as for module-to-module ir-
regularities. The relative response Rrel is a scale factor that
corrects forward relative to central jets and CMI is a cor-
rection for multiple interactions in the same bunch cross-
ing. The function CUE is a correction for the jet energy
added from the underlying event. D0 has one offset cor-
rection, CMI;UE, which includes the effects of multiple

interactions, the underlying event, noise from radioactive
decays of the uranium absorber, and the effect of collisions
from previous bunch crossings (pileup). The functions COC

and FOC are corrections for shower particles scattered in or
out of the cone of radiusR. CDF’s correction accounts for
MC modeling that affects how the parton energy is trans-
lated into particle jet energy, whereas D0’s correction
accounts for a detector effect caused by the finite cell
size in the calorimeter coupled with the cone size for the
jet algorithm. The combined jet energy scale corrections
increase the measured jet energies by about 20%–50%,
depending on pT and �.

The overall uncertainties on the jet energy scale correc-
tions vary from about 2.7% for CDF and 1.1% for D0 for
central jets of transverse energy of 100 GeV to 3.3% for
CDF and 2.2% for D0 for forward jets. Central jets of
25 GeV have correction uncertainties of 5.9% for CDF
and 1.4% for D0. For both experiments, the uncertainty on
the corrections for absolute response Rabs dominate these
uncertainties.

At D0, the jet energy resolution in data is lower than
predicted by the detector simulation. Therefore, the ener-
gies of MC jets are smeared so that the resulting resolution
in MC matches that in data. Similarly, the reconstruction
efficiency for jets in data is lower than is predicted by the

detector simulation, so an appropriate fraction of MC jets
are randomly removed. Both effects are corrected for as
functions of jet pT and pseudorapidity.
D0 Run II analyses include an energy correction to

simulated jets that depends on jet flavor. There are correc-
tions for b jets, other-quark flavor jets (u, d, s, and c), and
gluon jets implemented in both the leptonþ jets and di-
lepton analyses. Such corrections refine the simulation by
improving the matching of jet energies in MC to data. The
differences arise from the varying electromagnetic frac-
tions and widths of the jets. The corrections depend on jet
transverse energy and pseudorapidity and range from�6%
to þ2% [25].
Both collaborations perform an in situ jet energy scale

calibration in leptonþ jets events for the matrix-element
mass extraction of mt, and in CDF’s alljets and 6ET þ jets
measurements of mt. The invariant mass of the two jets is
constrained to a Breit-Wigner distribution for theW ! q �q0
decay, set to the world average value for theW-boson mass
[7]. The energies of all jets in the event are then rescaled to
complete this calibration.

F. b-quark jet properties

To separate top-quark events from background and to
decrease the ambiguity in jet-to-parton matching, it is
important to identify b-quark jets. Every t�t event has two
b jets, whereas such jets are rare in background. As B
hadrons have a mean lifetime of � 10�12 s, b jets can be
tagged through secondary vertices of the B decay a few
mm away from the primary p �p interaction. CDF’s
b-tagging algorithm uses the significance of the displace-
ment of the secondary vertex in the transverse ðr; �Þ plane
for the leptonþ jets and 6ET þ jets channels [57], as well as
a jet-probability algorithm for 6ET þ jets events [83]. One
parameter defines the significance of the separation of the
primary and secondary vertices for events with one and two
b jets. For jets that are within the acceptance of the silicon
microstrip tracker (i.e., ‘‘taggable’’ jets), this algorithm
identifies 50% of real b jets and 9% of real charm jets,
while falsely tagging 1% of light jets. D0 tags jets by
combining nine track and secondary-vertex-related varia-
bles using a neural network [60]. For jets within the
acceptance of the silicon microstrip detector, this yields
efficiencies of 65% and 20% for real b and charm jets,
respectively, while falsely tagging 3% of light jets.
To identify heavy-flavor jets in data and in MC events,

the tagging algorithm is applied by CDF and D0 directly to
the jets, except for simulated W þ light jets events, where
CDF uses tag-rate functions measured in multijet data,
since the rate for directly tagged MC events is very low.
After applying direct tagging to b and c jets in MC events,
D0 corrects the tagging efficiencies to match those ob-
served in data by randomly dropping the tagging of 13%
of such jets. For light-flavor jets, D0 assigns a per jet
mistag weight.

T. AALTONEN et al. PHYSICAL REVIEW D 86, 092003 (2012)

092003-12



G. Properties of other event observables

The uncertainty on mt depends not only on an accurate
measurement of jet energies and proper assignment of
flavor but also on the reconstruction and calibration of
the other elements of the event, including electrons,
muons, and the imbalance in transverse momentum, taking
into account the presence of any simultaneous p �p inter-
actions in the same bunch crossing.

The mean number of p �p collisions per bunch crossing is
� 2 in Run I and� 5 in Run II. Such additional collisions
affect the observed characteristics of the hard scatter of
interest and must be included in the MC simulation.
These extra collisions result mostly in the production of
low-pT particles. CDF simulates such additional interac-
tions using the PYTHIA model of minimum-bias events and
overlays them onto the hard scatters using a Poisson mean
appropriate to the instantaneous luminosity of the data. In a
similar manner D0 overlays randomly triggered data events
with the same luminosity profile as the data onto the MC
simulated events.

Electrons are identified by matching clusters of energy
deposited in the electromagnetic layers of the calorimeters
with tracks that point from the primary collision vertex to
the clusters. The spatial shapes of the showers must agree
with those expected for electrons, as studied in test-beam
data. The energy of an electron is determined as a combi-
nation of the total energy of the cluster and the momentum
measured from the curvature of the matching track. The
reconstruction efficiency is determined using Z ! ee data
by identifying one tight charged lepton as a tag and using the
other charged lepton as a probe (tag-and-probe method).
The electron energy is also recalibrated using suchZ events.

Muons are reconstructed from a central track and
matched to a track in the outer muon chambers. In D0,
both the inner and outer trajectories pass through magnetic
fields, and so the transverse momenta of the two are there-
fore required to match. The reconstruction efficiency and
calibration of pT are determined using a tag-and-probe
method applied on J=c ! �� and Z ! �� events in a
manner similar to that used for electrons.

As indicated above, all t�t decay channels except for
alljets events have a large 6ET . All jet energy calibration
corrections are also propagated to 6ET in each event.

III. COMBINATION OF MASS MEASUREMENTS

A. BLUE combination method

The basic idea of the technique, called the best linear
unbiased estimator (BLUE) method [32–34], used to ob-
tain the combined mass mcomb

t , an ‘‘estimator’’ of the true
mass mtrue

t , is to calculate a linear weighted sum of the
results from separate measurements:

mcomb
t ¼ X12

i¼1

wim
i
t: (1)

The mi
t are the 12 CDF and D0 measurements i of mt and

X12

i¼1

wi ¼ 1: (2)

The weights are determined using the value of mcomb
t that

minimizes the squared difference relative to the unknown
true value mtrue

t :

ðmcomb
t �mtrue

t Þ2¼Varianceðmcomb
t Þþ½Biasðmcomb

t Þ�2; (3)

where the two terms represent the weighted variance and
bias in the 12 input mt values with

Varianceðmcomb
t Þ ¼ X12

i¼1

w2
iVarianceðmi

tÞ; (4)

and

Varianceðmi
tÞ ¼ ½�ðmi

t�2; (5)

where �ðmi
tÞ are the uncertainties on the 12 input values

given in Table I.
On average, we expect the input mass measurements to

be unbiased, and we therefore assume

Biasðmcomb
t Þ ¼ X12

i¼1

wiBiasðmi
tÞ ¼ 0: (6)

Equation (3) shows that the BLUE method defines the best
estimate through a minimization of the variance of mt for
an assumed unbiased set of measurements. The minimum
corresponds to setting the weights to

wi ¼ 1=Varianceðmi
tÞP

12
i¼1 1=Varianceðmi

tÞ
(7)

for uncorrelated input values. Since the inputmt values are
correlated, the variance in Eq. (4) has to be replaced with a
covariance matrix:

Varianceðmcomb
t Þ ¼ X12

i¼1

X12

j¼1

wiwjCovarianceðmi
t; m

j
t Þ; (8)

which is defined as

Covarianceðmi
t; m

j
t Þ ¼ ½�ðmi

tm
j
t Þ�2 � �ðmi

tÞ�ðmj
t Þ: (9)

Minimizing Eq. (3) yields

wi ¼
P

12
j¼1 Covariance

�1ðmi
t; m

j
t Þ

P
12
i¼1

P
12
j¼1 Covariance

�1ðmi
t; m

j
t Þ
; (10)

where Covariance�1ðmi
t; m

j
t Þ are the elements of the in-

verse of the covariance matrix (also known as the error
matrix), and

Covarianceðmi
t;m

j
t Þ¼Correlationðmi

t;m
j
t Þ�ðmi

tÞ�ðmj
t Þ

(11)

with Correlationðmi
t; m

j
t Þ the correlation coefficient be-

tween mi
t and mj

t . The following sections show how the

COMBINATION OF THE TOP-QUARK MASS . . . PHYSICAL REVIEW D 86, 092003 (2012)

092003-13



correlation matrix is derived by examining the uncertainty
components and their individual correlations.

B. Measurement uncertainties

The uncertainty on any mt measurement has a statistical
component from the limited number of events available for
the measurement and a systematic component from the
uncertainties assigned to the calibration of input quantities,
to the model of the signal, and to the calibration of the mass
extraction method. Since the first measurements of mt

[4,5], the systematic component has been slightly larger
than the statistical one. As more data became available, the
statistical uncertainties on mt improved as did the calibra-
tions of systematic uncertainty, and the two components
therefore improved together.

The systematic uncertainty on each mt measurement in
this combination is divided into 14 parts. Some of them
have origin in only one source, whereas others include
several related sources of uncertainties. For the latter the
patterns of correlation among different channels, Tevatron
Run I and Run II, or experiments are the same for all
sources included in these systematic components. The
uncertainty on jet energy scale (JES), on the other hand,
is split into seven components, which do not apply to all
measurements, given the significantly different approaches
to jet energy calibration between CDF and D0 and the
change in the D0 procedure between Run I and Run II.

Table II gives the uncertainty of each of the 12 top-quark
mass measurements for the different contributions to
uncertainty and their effect on the final combination. The
components of uncertainty are defined in the following and
can be classified as uncertainties in detector response (jet
energy scale, jet and lepton modeling), uncertainties from
modeling signal and background (signal modeling, mul-
tiple interactions model, background estimated from the-
ory, and background based on data), uncertainties from
method of mass extraction, and statistical uncertainties.
A detailed description of the methods to evaluate these
systematic uncertainties is presented in the Appendix.

1. Jet energy scale

a. Light-jet response (1)

One subcomponent of the uncertainty in JES covers the
absolute calibration for CDF’s Run I and Run II measure-
ments. It also includes small contributions from the un-
certainties associated with modeling multiple interactions
within a single bunch crossing and corrections for the
underlying event.

b. Light-jet response (2)

Another subcomponent of this uncertainty includes D0’s
Run I and Run II calibrations of absolute response (energy
dependent), the relative response (� dependent), and the

TABLE II. The uncertainty in GeV from each component for the 12 measurements of mt and the resulting Tevatron combination.
The total uncertainties are obtained by adding the components in quadrature. The entries ‘‘n/a’’ stand for ‘‘not applicable’’ and ‘‘n/e’’
for ‘‘not evaluated.’’ The nonevaluated uncertainties were not considered as significant sources of uncertainty for Run I measurements.

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Channel Run Expt. Jet energy scale systematics Other systematics

Leptonþ jets II CDF 0.41 0.01 0.27 n/a 0.23 0.13 0.58 0.00 0.14 0.56 0.10 0.27 0.06 0.10 0.65 0.80 0.67 1.23

Leptonþ jets II D0 n/a 0.63 n/a n/a 0.07 0.26 0.46 0.36 0.18 0.77 0.05 0.19 0.23 0.16 0.83 0.83 0.94 1.50

Leptonþ jets I CDF 3.4 0.7 2.7 n/a 0.6 n/e n/a n/e n/e 2.7 n/e 1.3 n/e 0.0 5.1 4.4 2.8 7.3

Leptonþ jets I D0 n/a 2.5 2.0 1.3 0.7 n/e n/a n/e n/e 1.3 n/e 1.0 n/e 0.6 3.6 3.5 1.6 5.3

Alljets II CDF 0.38 0.04 0.24 n/a 0.15 0.03 0.95 0.00 n/a 0.64 0.08 0.00 0.56 0.38 1.43 1.06 0.91 2.00

Alljets I CDF 4.0 0.3 3.0 n/a 0.6 n/e n/a n/e n/a 2.1 n/e 1.7 n/e 0.6 10.0 5.0 2.6 11.5

Dileptons II CDF 2.01 0.58 2.13 n/a 0.33 0.14 n/a 0.00 0.27 0.80 0.23 0.24 0.14 0.12 1.95 3.01 0.88 3.69

Dileptons II D0 n/a 0.56 n/a n/a 0.20 0.40 0.55 0.50 0.35 0.86 0.00 0.00 0.20 0.51 2.36 0.90 1.11 2.76

Dileptons I CDF 2.7 0.6 2.6 n/a 0.8 n/e n/a n/e n/e 3.0 n/e 0.3 n/e 0.7 10.3 3.9 3.0 11.4

Dileptons I D0 n/a 1.1 2.0 1.3 0.7 n/e n/a n/e n/e 1.9 n/e 1.1 n/e 1.1 12.3 2.7 2.3 12.8

6ET þ jets II CDF 0.45 0.05 0.20 n/a 0.00 0.12 1.54 0.00 n/a 0.78 0.16 0.00 0.12 0.14 1.80 1.64 0.78 2.56

Decay length II CDF 0.24 0.06 n/a n/a 0.15 n/e n/a 0.00 n/a 0.90 0.00 0.80 0.20 2.50 9.00 0.25 2.80 9.43

Tevatron combination 0.12 0.19 0.04 0.00 0.15 0.12 0.39 0.11 0.10 0.51 0.00 0.14 0.11 0.09 0.56 0.49 0.57 0.94

T. AALTONEN et al. PHYSICAL REVIEW D 86, 092003 (2012)

092003-14



out-of-cone showering correction that is a detector effect.
This uncertainty term for CDF includes only the small
relative response calibration (� dependent) for Run I and
Run II.

c. Out-of-cone correction

This subcomponent of the JES uncertainty quantifies the
out-of-cone showering corrections to theMC showers for all
of CDF’s and for D0’s Run Imeasurements that are obtained
by varying the model for light-quark fragmentation.

d. Offset

This subcomponent originates from the offset in D0’s
Run I calibration, which corrects for noise from uranium
decay, pileup from previous collisions, and for multiple
interactions and the model for the underlying event. In
Run I, the uncertainties are large, but in Run II, owing to
the smaller integration time for calorimeter electronics,
they are negligible. CDF’s calorimeter does not have the
same sources of noise and sensitivity to pileup as D0, so
CDF measurements do not have this term.

e. Model for b jets

This subcomponent comes from the uncertainty on the
semileptonic branching fraction in b decays and from
differences between two models of b-jet hadronization.

f. Response to b=q=g jets

This subcomponent accounts for the difference in the
electromagnetic versus hadronic response of b jets, light-
quark jets, and gluon jets. CDF corrects for jet flavor as part
of the main calibration, and defines the uncertainty based
on the remaining difference in response between b jets and
light-flavor jets, whereas D0 corrects the response for b,
light-quark (u, d, s, and c), and gluon jets as a function of
jet pT and �.

g. In situ light-jet calibration

The last part of the uncertainty in the jet energy scale is
from the in situ calibration of mt. It corresponds to the
statistical uncertainty from the limited number of events
used in the fit when using the W-boson mass to constrain
the energies of the light quarks from the W decay.

2. Jet modeling

The uncertainty in jet modeling has two components for
D0. This uncertainty is negligible for CDF.

(i) The jet energy resolution is smeared for MC jets to
match the resolution observed in data, and the uncer-
tainty on the smearing functions is propagated to mt.

(ii) The identification efficiency in MC events is cor-
rected tomatch that found in data, and the uncertainty
on the correction functions is propagated to mt.

3. Lepton modeling

This uncertainty has two components:
(i) The electron and muon pT scales are calibrated to

the J=c and Z-boson mass by both CDF and D0.
This uncertainty on the calibration is included in the
measurements of mt.

(ii) D0 smears the muon momentum resolution in MC
events to match that in data, and the uncertainty on
this correction is included in this term. The uncer-
tainty on the electron resolution has a negligible
impact on the measurements of mt.

4. Signal modeling

There are six components to this uncertainty. They are
combined into one term because the correlations between
channels are similar for each component:
(i) Knowledge of the PDF parametrization.
(ii) The quark annihilation and gluon fusion fractions

that differ significantly between leading-log and
next-to-leading-order (NLO) QCD calculations
(Run II).

(iii) The amount of initial- and final-state radiation in
MC signal events differs from that in data and is
adjusted through the value of �QCD used in the

shower and the scales of time and spacelike
showers.

(iv) Higher-order QCD corrections to initial- and final-
state radiation differ from precise parton-level
models, and this is not accounted for by the choice
of scale for the calculations (Run II).

(v) Our model for jet hadronization is based on angu-
lar ordering in PYTHIA with Tune A underlying-
event tuning. Parton showering and the underlying
event can also be simulated with HERWIG and
JIMMY [84,85]. The effect of the difference on

mt between the two models is included in this
term.

(vi) Final-state partons and remnants of the protons and
antiprotons are connected through color strings,
which affect the distributions of jets. Since this
effect is not included in the model for the t�t signal,
the value of mt has an uncertainty from this omis-
sion (Run II).

5. Multiple interactions model

The number of soft p �p events overlaid on each MC
event has a Poisson distribution. The mean number does
not equal exactly the number seen in data since the
luminosity increased as the Tevatron run progressed.
The top-quark mass is measured as a function of the
number of multiple interactions in signal events by
CDF, the signal MC events are reweighted to match the
distribution seen in data by D0, and the related uncertain-
ties are included here.

COMBINATION OF THE TOP-QUARK MASS . . . PHYSICAL REVIEW D 86, 092003 (2012)

092003-15



6. Background from theory

There are four components in this uncertainty:
(i) Difference between NLO calculations of the fraction

of heavy-flavor jets in W þ jets events. The ALPGEN

model underestimates this fraction.
(ii) Impact of factorization and renormalization scales

on the W þ jets simulation, which affects the back-
ground model for distributions characterizing jets.

(iii) The theoretical cross sections used to normalize all
MC estimated background processes (except for
W þ jets for CDF and D0 leptonþ jets measure-
ments, and Drell-Yan production for CDF dilepton
measurements).

(iv) Impact of difference between the MC modeling
of background kinematic distributions and those
observed in data.

7. Background based on data

This refers primarily to uncertainties from the normal-
ization of certain background components to data. These
include multijet backgrounds in the leptonþ jets, alljets,
and 6ET þ jets analyses, theW þ jets background in the D0
leptonþ jets analyses, and the Drell-Yan backgrounds in
the CDF dilepton analyses.

D0 also considers the following four components of
uncertainty:

(i) The uncertainty from correcting the MC events to
match the trigger efficiency in data, which is based
on the turn-on response for each trigger element.

(ii) The uncertainty from applying tag-rate and tagg-
ability corrections to MC events to make the effi-
ciencies match the data for each jet flavor.

(iii) The uncertainty on the fraction of multijet
events included in the pseudoexperiments used for
calibration.

8. Calibration method

The extracted values of mt are calibrated using a
straight-line fit to the relationship between input mass
and measured mass in simulated pseudoexperiments.
This term includes the systematic uncertainties from the
slope and offset of this calibration.

9. Statistical uncertainty

The statistical uncertainties are determined from the
number of data events in each of the 12 measurements.

Figure 5 shows the relative contribution for each major
uncertainty to the analysis channels in Run II. The
Appendix provides more detail on how each of the sources
of the uncertainties is estimated.

C. Uncertainty correlations

Tables III and IV indicate how uncertainties are corre-
lated between measurements. There are seven patterns of
correlation:

(i) Statistical uncertainty and calibration method uncer-
tainty are not correlated among the measurements.

(ii) Correlations among D0 measurements that imple-
ment the same final jet energy corrections for the
uncertainty from in situ light-jet calibration.

(iii) Correlations among CDF measurements that use
the same data samples for the uncertainty from
background based on data.

(iv) Correlations among all measurements in the same t�t
decay channel for the uncertainty from background
estimated from theory.

(v) Correlations of measurements within the same ex-
periment for a given run period for the uncertainties
from light-jet response (2), offset, response to
b=q=g jets, jet modeling, lepton modeling, and mul-
tiple interactions model.

(vi) Correlations for measurements within the same
experiment such as the uncertainty from light-jet
response (1).

(vii) Correlations among all measurements such as the
uncertainties from out-of-cone correction, model
for b jets, and signal modeling.

We assume that all sources correspond to either no or
100% correlation. A check of this assumption (see
Sec. IVB) shows that it has a negligible effect on the
combined value and uncertainty of mt.

D. Measurement correlations

The uncertainties shown in Table II and their correla-
tions shown in Tables III and IV provide the correlations
among the 12 input values ofmt. The correlation matrix for
these measurements, as returned by the combination pro-
cedure, is shown in Table V. The inversion of the covari-
ance matrix built with the correlation matrix defines the
measurement weights, as described in Sec. III A.

100%

90%

80%

70%

60%

50%

40%

30%

20%

10%

0%

%
 o

f 
th

e 
T

o
ta

l U
n

ce
rt

ai
n

ty
o

f 
th

e 
T

o
p

 Q
u

ar
k 

M
as

s 

Channels

Statistics

Method of mass extraction

Modeling background

Modeling signal

Jet energy scale

Source of uncertainty

Le
pt

on
+je

ts

Allje
ts

Dile
pt

on
s

ET
+je

ts

Dec
ay

 le
ng

th

Com
bin

at
ion

FIG. 5 (color online). The average uncertainties for CDF and
D0 for each Run II measurement and for the Tevatron combina-
tion, separated according tomajor components. (See Table VIII in
the Appendix for details on the systematic categories. In this
figure, the jet and lepton modeling systematic uncertainties are
grouped into the modeling background category.)

T. AALTONEN et al. PHYSICAL REVIEW D 86, 092003 (2012)

092003-16



TABLE III. Correlations in systematic uncertainties (in percent) among the different measurements of mt.

L
ep
to
n
+
je
ts

R
u
n
II

C
D
F

L
ep
to
n
+
je
ts

R
u
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D
0

L
ep
to
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+
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ts

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F

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ep
to
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+
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0

A
ll
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R
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A
ll
je
ts
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il
ep
to
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s
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il
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to
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0

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il
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to
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il
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to
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s
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u
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0

6ET
+
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ts

R
u
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II

C
D
F

D
ec
ay

le
n
g
th

R
u
n
II

C
D
F

Calibration method Statistical uncertainty

Not correlated among any measurements

In situ light-jet calibration (JES)

Leptonþ jets Run II CDF 100 0 0 0 0 0 0 0 0 0 0 0

Leptonþ jets Run II D0 0 100 0 0 0 0 0 100 0 0 0 0

Leptonþ jets Run I CDF 0 0 100 0 0 0 0 0 0 0 0 0

Leptonþ jets Run I D0 0 0 0 100 0 0 0 0 0 0 0 0

Alljets Run II CDF 0 0 0 0 100 0 0 0 0 0 0 0

Alljets Run I CDF 0 0 0 0 0 100 0 0 0 0 0 0

Dileptons Run II CDF 0 0 0 0 0 0 100 0 0 0 0 0

Dileptons Run II D0 0 100 0 0 0 0 0 100 0 0 0 0

Dileptons Run I CDF 0 0 0 0 0 0 0 0 100 0 0 0

Dileptons Run I D0 0 0 0 0 0 0 0 0 0 100 0 0

6ET þ jets Run II CDF 0 0 0 0 0 0 0 0 0 0 100 0

Decay length Run II CDF 0 0 0 0 0 0 0 0 0 0 0 100

Background based on data

Leptonþ jets Run II CDF 100 0 0 0 0 0 0 0 0 0 0 100

Leptonþ jets Run II D0 0 100 0 0 0 0 0 0 0 0 0 0

Leptonþ jets Run I CDF 0 0 100 0 0 0 0 0 0 0 0 0

Leptonþ jets Run I D0 0 0 0 100 0 0 0 0 0 0 0 0

Alljets Run II CDF 0 0 0 0 100 0 0 0 0 0 0 0

Alljets Run I CDF 0 0 0 0 0 100 0 0 0 0 0 0

Dileptons Run II CDF 0 0 0 0 0 0 100 0 0 0 0 0

Dileptons Run II D0 0 0 0 0 0 0 0 100 0 0 0 0

Dileptons Run I CDF 0 0 0 0 0 0 0 0 100 0 0 0

Dileptons Run I D0 0 0 0 0 0 0 0 0 0 100 0 0

6ET þ jets Run II CDF 0 0 0 0 0 0 0 0 0 0 100 0

Decay length Run II CDF 100 0 0 0 0 0 0 0 0 0 0 100

Background from theory

Leptonþ jets Run II CDF 100 100 100 100 0 0 0 0 0 0 0 100

Leptonþ jets Run II D0 100 100 100 100 0 0 0 0 0 0 0 100

Leptonþ jets Run I CDF 100 100 100 100 0 0 0 0 0 0 0 100

Leptonþ jets Run I D0 100 100 100 100 0 0 0 0 0 0 0 100

Alljets Run II CDF 0 0 0 0 100 100 0 0 0 0 0 0

Alljets Run I CDF 0 0 0 0 100 100 0 0 0 0 0 0

Dileptons Run II CDF 0 0 0 0 0 0 100 100 100 100 0 0

Dileptons Run II D0 0 0 0 0 0 0 100 100 100 100 0 0

Dileptons Run I CDF 0 0 0 0 0 0 100 100 100 100 0 0

Dileptons Run I D0 0 0 0 0 0 0 100 100 100 100 0 0

6ET þ jets Run II CDF 0 0 0 0 0 0 0 0 0 0 100 0

Decay length Run II CDF 100 100 100 100 0 0 0 0 0 0 0 100

Light-jet response (2) (JES) Offset (JES) Response to b=q=g jets (JES)

Jet modeling Lepton modeling Multiple interactions model

Leptonþ jets Run II CDF 100 0 0 0 100 0 100 0 0 0 100 100

Leptonþ jets Run II D0 0 100 0 0 0 0 0 100 0 0 0 0

Leptonþ jets Run I CDF 0 0 100 0 0 100 0 0 100 0 0 0

COMBINATION OF THE TOP-QUARK MASS . . . PHYSICAL REVIEW D 86, 092003 (2012)

092003-17



E. Measurement weights

As discussed in Sec. III A, the combined mass mcomb
t is

defined through the set of weights that minimize the
squared difference between mcomb

t and the true value of
mt, which is equivalent to minimizing the sum of the
covariance matrix elements. Table V gives the weights wi

for each of the input measurements as determined in this
minimization. A weight of zero means that an input mea-
surement has no effect on mcomb

t . The Run I measurement
weights are negative, which reflects the fact that the corre-
lations for these and other measurements are larger than the
ratio of their total uncertainties [33]. In this case, the less
precise measurement may acquire a negative weight. Input
measurements with negative weights still affect the value
of mcomb

t and reduce the total uncertainty. By design, the
sum of the weights is set to unity.

IV. RESULTS OF THE COMBINATION

A. Tevatron top-quark mass result

Combining the 12 independent measurements of mt

from the CDF and D0 Collaborations yields

mcomb
t ¼ 173:18� 0:56 ðstatÞ � 0:75 ðsystÞ GeV

¼ 173:18� 0:94 GeV:

The uncertainties are split into their components in Table II
and Fig. 5. The jet energy scale contributes 0.49 GeV to the
total systematic uncertainty. Of this, 0.39 GeV arises from
limited statistics of the in situ JES calibration and 0.30 GeV

from the remaining contributions. Figure 6 summarizes the
input mt values and the combined result.
We assess the consistency of the input mt measurements

with their combination using a 	2 test statistic, defined as
follows:

	2
comb ¼ ðmi

t �mcomb
t ÞTCovariance�1

� ðmi
t; m

j
t Þðmj

t �mcomb
t Þ;

wheremi
t is a column vector of the 12mt inputs,m

comb
t is a

matching column vector for the measurements adjusted in
the previous minimization, and the superscript T denotes
the transpose. We find

	2
comb ¼ 8:3 for 11 degrees of freedom;

which is equivalent to a 69% probability for agreement
(i.e., p value for the observed 	2 value) among the 12 input
measurements.

B. Consistency checks

We check one aspect of the assumption that biases in the
inputmt are on average zero (see Sec. III A) by calculating
separately the combined mcomb

t for each t�t decay mode,
each run period, and each experiment. The results are
shown in Table VI. The resulting mcomb

t values are calcu-
lated using all 12 input measurements and their correla-
tions. The 	2 test statistic provides the compatibility of
each subset with the others and is defined as

	2
sub1;sub2;¼ ðmsub1

t �msub2
t Þ2Covariance�1ðmsub1

t �msub2
t Þ:

L
ep
to
n
+
je
ts

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C
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ep
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+
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ts

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II

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+
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ts

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C
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ep
to
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+
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ts
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D
0

A
ll
je
ts

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u
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C
D
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A
ll
je
ts
R
u
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C
D
F

D
il
ep
to
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s
R
u
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II

C
D
F

D
il
ep
to
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s
R
u
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II

D
0

D
il
ep
to
n
s
R
u
n
I
C
D
F

D
il
ep
to
n
s
R
u
n
I
D
0

6ET
+
je
ts
R
u
n
II

C
D
F

D
ec
ay

le
n
g
th

R
u
n
II

C
D
F

Calibration method Statistical uncertainty

Not correlated among any measurements

In situ light-jet calibration (JES)

Leptonþ jets Run I D0 0 0 0 100 0 0 0 0 0 100 0 0

Alljets Run II CDF 100 0 0 0 100 0 100 0 0 0 100 100

Alljets Run I CDF 0 0 100 0 0 100 0 0 100 0 0 0

Dileptons Run II CDF 100 0 0 0 100 0 100 0 0 0 100 100

Dileptons Run II D0 0 100 0 0 0 0 0 100 0 0 0 0

Dileptons Run I CDF 0 0 100 0 0 100 0 0 100 0 0 0

Dileptons Run I D0 0 0 0 100 0 0 0 0 0 100 0 0

6ET þ jets Run II CDF 100 0 0 0 100 0 100 0 0 0 100 100

Decay length Run II CDF 100 0 0 0 100 0 100 0 0 0 100 100

TABLE III. (Continued)

T. AALTONEN et al. PHYSICAL REVIEW D 86, 092003 (2012)

092003-18



TABLE V. Correlations in % among the input mt measurements and their weights in the BLUE combination.

Leptonþ jets

Run II CDF

Leptonþ jets

Run II D0

Leptonþ jets

Run I CDF

Leptonþ jets

Run I D0

Alljets

Run II CDF

Alljets

Run I CDF

Dileptons

Run II CDF

Dileptons

Run II D0

Dileptons

Run I CDF

Dileptons

Run I D0

6ET þ jets

Run II CDF

Decay length

Run II CDF

Weight

Leptonþ jets Run II CDF 100 27 45 25 25 26 44 12 26 11 24 8 55.50

Leptonþ jets Run II D0 27 100 21 14 16 9 11 39 13 7 15 6 26.66

Leptonþ jets Run I CDF 45 21 100 26 25 32 54 12 29 11 22 7 �4:72

Leptonþ jets Run I D0 25 14 26 100 12 14 27 7 15 16 10 5 �0:06

Alljets Run II CDF 25 16 25 12 100 15 25 10 15 7 14 4 13.99

Alljets Run I CDF 26 9 32 14 15 100 38 6 19 7 14 4 �0:80

Dileptons Run II CDF 44 11 54 27 25 38 100 7 32 13 22 6 1.41

Dileptons Run II D0 12 39 12 7 10 6 7 100 8 5 10 3 2.28

Dileptons Run I CDF 26 13 29 15 15 19 32 8 100 8 14 4 �1:05

Dileptons Run I D0 11 7 11 16 7 7 13 5 8 100 6 2 �0:15

6ET þ jets Run II CDF 24 15 22 10 14 14 22 10 14 6 100 4 6.65

Decay length Run II CDF 8 6 7 5 4 4 6 3 4 2 4 100 0.29

TABLE IV. Correlations in systematic uncertainties (in percent) among the different measurements of mt (continued).

Leptonþ jets

Run II CDF

Leptonþ jets

Run II D0

Leptonþ jets

Run I CDF

Leptonþ jets

Run I D0

Alljets

Run II CDF

Alljets

Run I CDF

Dileptons

Run II CDF

Dileptons

Run II D0

Dileptons

Run I CDF

Dileptons

Run I D0

6ET þ jets

Run II CDF

Decay length

Run II CDF

Light-jet response (1) (JES)

Leptonþ jets Run II CDF 100 0 100 0 100 100 100 0 100 0 100 100

Leptonþ jets Run II D0 0 100 0 100 0 0 0 100 0 100 0 0

Leptonþ jets Run I CDF 100 0 100 0 100 100 100 0 100 0 100 100

Leptonþ jets Run I D0 0 100 0 100 0 0 0 100 0 100 0 0

Alljets Run II CDF 100 0 100 0 100 100 100 0 100 0 100 100

Alljets Run I CDF 100 0 100 0 100 100 100 0 100 0 100 100

Dileptons Run II CDF 100 0 100 0 100 100 100 0 100 0 100 100

Dileptons Run II D0 0 100 0 100 0 0 0 100 0 100 0 0

Dileptons Run I CDF 100 0 100 0 100 100 100 0 100 0 100 100

Dileptons Run I D0 0 100 0 100 0 0 0 100 0 100 0 0

6ET þ jets Run II CDF 100 0 100 0 100 100 100 0 100 0 100 100

Decay length Run II CDF 100 0 100 0 100 100 100 0 100 0 100 100

Out-of-cone correction (JES) Model for b jets (JES) Signal modeling

100% correlated among all measurements

C
O
M
B
IN

A
T
IO

N
O
F
T
H
E
T
O
P
-Q

U
A
R
K

M
A
S
S
...

P
H
Y
S
IC
A
L
R
E
V
IE
W

D
8
6
,
0
9
2
0
0
3
(2
0
1
2
)

0
9
2
0
0
3
-1
9



The 	2 values in Table VI show that biases in the input
measurements are not large.

To check the impact of the assumption that the system-
atic uncertainty terms are either 0% or 100% correlated
between input measurements, we change all off-diagonal
100% values to 50% (see Tables III and IV) and recalculate
the combined top-quark mass. This extreme change shifts
the central mass value up by 0.17 GeV and reduces the
uncertainty negligibly. The chosen approach is therefore
conservative.

C. Summary

We have combined 12 measurements of the mass of the
top quark by the CDF and D0 collaborations at the
Tevatron collider and find

mcomb
t ¼ 173:18� 0:56 ðstatÞ � 0:75 ðsystÞ GeV;

which corresponds to a precision of 0.54%. The result is
shown in Table VII together with previous combined re-
sults for comparison. The input measurements for this
combination use up to 5:8 fb�1 of integrated luminosity
for each experiment, while 10 fb�1 are now available. We
therefore expect the final combination to improve in pre-
cision with the use of all the data, but also from analyzing
all t�t decay channels in both experiments and from the
application of improved measurement techniques, signal
and background models, and calibration corrections to
all channels that will reduce systematic uncertainties.
Currently, there are also some overlaps of the systematic
effects that are included in different uncertainty categories.

TABLE VI. Separate calculations of mcomb
t for each t�t decay mode, by run period, and by experiment, and their 	2 probabilities.

Subset mcomb
t Consistency 	2 (Degrees of freedom ¼ 1) 	2 probability

Leptonþ
jets Alljets Dileptons

6ETþ
jets

Run

II-Run I CDF-D0

Leptonþ
jets Alljets Dileptons

6ETþ
jets

Run

II-Run I CDF-D0

Leptonþ jets 173:4� 1:0 � � � 0.14 1.51 0.28 � � � 71% 22% 60%

Alljets 172:7� 1:9 0.14 � � � 0.40 0.04 71% � � � 53% 85%

Dileptons 171:1� 2:1 1.51 0.40 � � � 0.12 22% 53% � � � 73%

6ET þ jets 172:1� 2:5 0.28 0.04 0.12 � � � 60% 85% 73% � � �
Run II 173:6� 1:0 2.89 9%

Run I 180:0� 4:1

CDF 172:5� 1:0 2.56 11%

D0 174:9� 1:4

160 170 180 190

Mass of the Top Quark   [GeV] 

Lepton+jets

Lepton+jets

Lepton+jets

Lepton+jets

Alljets

Alljets

Dileptons

Dileptons

Dileptons

Dileptons

ET +jets

Decay length

Tevatron Combination 2012

Run II

Run II

Run I

Run I

Run II

Run I

Run II

Run II

Run I

Run I

Run II

Run II

CDF

D

CDF

D

CDF

CDF

CDF

D

CDF

D

CDF

CDF

1.06

1.24

5.3

3.9

1.40

5.7

3.13

1.44

4.9

3.6

1.82

2.82

0.75

  0.65

  0.83

  5.1

  3.6

  1.43

10.0

  1.95

  2.36

10.3

12.3

  1.80

  9.00

  0.56

173.00

174.94

176.1

180.1

172.47

186.0

170.28

174.00

167.4

168.4

172.32

166.90

173.18

GeV

GeV

GeV

GeV

GeV

GeV

GeV

GeV

GeV

GeV

GeV

GeV

GeV

FIG. 6 (color online). The 12 input measurements of mt from the Tevatron collider experiments along with the resulting combined
value of mcomb

t . The gray region corresponds to �0:94 GeV.

T. AALTONEN et al. PHYSICAL REVIEW D 86, 092003 (2012)

092003-20



In addition to the in situ light-jet calibration systematic
uncertainty that will scale down with the increase of ana-
lyzed luminosity, these levels of double counting are
expected to be reduced for the next combination. The
combination presented here has a 0.54% precision on mt,
making the top quark the particle with the best known mass
in the SM.

ACKNOWLEDGMENTS

We thank the Fermilab staff and technical staffs of the
participating institutions for their vital contributions and
acknowledge support from the DOE and NSF (USA), ARC
(Australia), CNPq, FAPERJ, FAPESP, and FUNDUNESP
(Brazil), NSERC (Canada), NSC, CAS, and CNSF
(China), Colciencias (Colombia), MSMT and GACR
(Czech Republic), the Academy of Finland, CEA, and
CNRS/IN2P3 (France), BMBF and DFG (Germany),
DAE and DST (India), SFI (Ireland), INFN (Italy),
MEXT (Japan), the Korean World Class University
Program and NRF (Korea), CONACyT (Mexico), FOM
(Netherlands), MON, NRC KI, and RFBR (Russia), the
Slovak R&D Agency, the Ministerio de Ciencia e
Innovación, and Programa Consolider-Ingenio 2010
(Spain), The Swedish Research Council (Sweden), SNSF
(Switzerland), STFC and the Royal Society (United
Kingdom), and the A. P. Sloan Foundation (USA).

APPENDIX A: EVALUATION OF SYSTEMATIC
UNCERTAINTIES

Systematic uncertainties arise from inadequate model-
ing of signal and backgrounds and from the inability to
reproduce the detector response with simulated events.
Systematic uncertainties also arise from ambiguities in
reconstructing the top quarks from their jet and lepton
remnants. We minimize such uncertainties by using inde-
pendent data to calibrate the absolute response of the
detector, and we use state-of-the-art input from theory for

modeling the signal and backgrounds. We use alternative
models for signal and different parameters for modeling
backgrounds to check our assumptions.
Table VIII lists the uncertainties from the Run II

leptonþ jets measurements for CDF and D0 that are based
on the matrix-element technique [24,25]. These two mea-
surements provide most of the sensitivity to the combined
mt result and are discussed below. Before explaining how
each individual systematic uncertainty is estimated, we
will first discuss how the uncertainties from different
sources are propagated to mt and how they are calculated
using ensembles of pseudoexperiments.
Uncertainties related to the performance of the detector

and calibration of the reconstructed objects, such as JES,
the modeling of jets, leptons, and triggers, and calibration
of the b-tagging algorithms, are evaluated by shifting the
central values of their respective parameters by �1 stan-
dard deviations (�) that correspond to the uncertainties on
each value. This is done using MC t�t events for mt ¼
172:5 GeV. The integrations over the matrix element are
performed again for each shifted sample and define shifts
in mt that correspond to each independent source of sys-
tematic uncertainty. These uncertainties are not determined
at other mt values, and it is assumed that their dependence
on mt is minimal.
For uncertainties that arise from ambiguities in the

modeling of the t�t signal, which include the uncertainties
from initial- and final-state radiation, higher-order QCD
corrections, b-jet hadronization, light-jet hadronization,
the underlying-event model, and color reconnection, we
generate simulated t�t events using alternative models also
at mt ¼ 172:5 GeV. These events are processed through
detector simulation and are reconstructed, and the proba-
bility density is calculated by integration over the matrix
elements.
For the uncertainties from the choice of parton distribu-

tion functions, the ratio of contribution from quark annihi-
lation and gluon fusion, and models for overlapping

TABLE VII. Mass measurements of the top quark from 1999 until this publication at the
Tevatron collider.

Year Integrated luminosity [fb�1] mt [GeV] Uncertainty on mt Reference

1999 0.1 174:3� 3:2� 4:0 2.9% [35]

2004 0.1 178:0� 2:7� 3:3 2.4% [36]

2005 0.3 172:7� 1:7� 2:4 1.7% [37]

2006 0.7 172:5� 1:3� 1:9 1.3% [38]

2006 1.0 171:4� 1:2� 1:8 1.2% [39]

2007 2.1 170:9� 1:1� 1:5 1.1% [40]

2008 2.1 172:6� 0:8� 1:1 0.8% [41]

2008 2.1 172:4� 0:7� 1:0 0.7% [42]

2009 3.6 173:1� 0:6� 1:1 0.7% [43]

2010 5.6 173:32� 0:56� 0:89 0.61% [44]

2011 5.8 173:18� 0:56� 0:75 0.54% [45]

5.8 173:18� 0:56� 0:75 0.54% This paper

COMBINATION OF THE TOP-QUARK MASS . . . PHYSICAL REVIEW D 86, 092003 (2012)

092003-21



interactions, we reweight the fully reconstructed simulated
t�tMC events atmt ¼ 165, 170, 172.5, 175, and 180 GeV to
reflect the uncertainty on the�1� range on each parameter
and extract its impact on mt.

Each method used to measure mt is calibrated using t�t
MC events generated at mt ¼ 165, 170, 172.5, 175,
180 GeV, which provide the relationship between input
and ‘‘measured’’ masses. A straight line is fitted to these

values, representing a response function that is used to
correct the mt measurement in data.
Systematic uncertainties are evaluated using studies of

ensembles of pseudoexperiments. For each of the shifted or
reweighted sets of events, and those based on alternative
models or different generatedmt, we create an ensemble of
at least 1000 pseudoexperiments, by means of binomially
smeared signal and background fractions that match the

TABLE VIII. Individual components of uncertainty on CDF and D0 mt measurements in the
leptonþ jets channel for Run II data [24,25].

Uncertainty [GeV]

Systematic CDF (5:6 fb�1) D0 (3:6 fb�1)

Source mt ¼ 173:00 GeV mt ¼ 174:94 GeV

DETECTOR RESPONSE

Jet energy scale

Light-jet response (1) 0.41 n/a

Light-jet response (2) 0.01 0.63

Out-of-cone correction 0.27 n/a

Model for b jets 0.23 0.07

Semileptonic b decay 0.16 0.04

b-jet hadronization 0.16 0.06

Response to b=q=g jets 0.13 0.26

In situ light-jet calibration 0.58 0.46

Jet modeling 0.00 0.36

Jet energy resolution 0.00 0.24

Jet identification 0.00 0.26

Lepton modeling 0.14 0.18

MODELING SIGNAL

Signal modeling 0.56 0.77

Parton distribution functions 0.14 0.24

Quark annihilation fraction 0.03 n/a

Initial and final-state radiation 0.15 0.26

Higher-order QCD corrections n/a 0.25

Jet hadronization and underlying event 0.25 0.58

Color reconnection 0.37 0.28

Multiple interactions model 0.10 0.05

MODELING BACKGROUND

Background from theory 0.27 0.19

Higher-order correction for heavy flavor 0.03 0.07

Factorization scale for W þ jets 0.07 0.16

Normalization to predicted cross sections 0.25 0.07

Distribution for background 0.07 0.03

Background based on data 0.06 0.23

Normalization to data 0.00 0.06

Trigger modeling 0.00 0.06

b-tagging modeling 0.00 0.10

Signal fraction for calibration n/a 0.10

Impact of multijet background on the calibration n/a 0.14

METHOD OF MASS EXTRACTION

Calibration method 0.10 0.16

STATISTICAL UNCERTAINTY 0.65 0.83

UNCERTAINTY ON JET ENERGY SCALE 0.80 0.83

OTHER SYSTEMATIC UNCERTAINTIES 0.67 0.94

TOTAL UNCERTAINTY 1.23 1.50

T. AALTONEN et al. PHYSICAL REVIEW D 86, 092003 (2012)

092003-22



expectation in the data sample and with the total number of
events in each pseudoexperiment equal to the number of
events observed in data. We use the ensembles of such
pseudoexperiments to assess the difference between gen-
erated and measured mass and to calibrate the method of
mass extraction.

For the uncertainty on background, we change the frac-
tion of background events in the pseudoexperiments within
their uncertainties and remeasure the top-quark mass.

For the BLUE combination method, the uncertainties
must be defined symmetrically around the central mass
value, and this requirement determines part of the follow-
ing definitions of uncertainty.

For the uncertainties obtained in ensemble studies with
shifted or reweighted parameters, mþ

t corresponds to the
þ1� shift in the input parameter and m�

t corresponds to
the �1� shift. The systematic uncertainty on the value of
mt from these parameters is defined as �jmþ

t �m�
t j=2,

unless both shifts are in the same direction relative to the
nominal value, in which case the systematic uncertainty is
defined as the larger of jmþ

t �mtj or jm�
t �mtj.

For the values obtained from a comparison between two
or more models, the systematic uncertainty is taken as� of
the largest difference among the resulting masses (without
dividing by two).

1. Jet energy scale

The following seven terms (1.1–1.7) refer to the jet
energy scale.

a. Light-jet response (1)

This uncertainty includes the absolute calib