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Measurement of the W-boson mass in pp collisions at \(\sqrt{s}=7\, \hbox {TeV}\) with the ATLAS detector

  • M. Aaboud
  • G. Aad
  • B. Abbott
  • J. Abdallah
  • O. Abdinov
  • B. Abeloos
  • S. H. Abidi
  • O. S. AbouZeid
  • N. L. Abraham
  • H. Abramowicz
  • H. Abreu
  • R. Abreu
  • Y. Abulaiti
  • B. S. Acharya
  • S. Adachi
  • L. Adamczyk
  • D. L. Adams
  • J. Adelman
  • M. Adersberger
  • T. Adye
  • A. A. Affolder
  • T. Agatonovic-Jovin
  • C. Agheorghiesei
  • J. A. Aguilar-Saavedra
  • S. P. Ahlen
  • F. Ahmadov
  • G. Aielli
  • S. Akatsuka
  • H. Akerstedt
  • T. P. A. Åkesson
  • A. V. Akimov
  • G. L. Alberghi
  • J. Albert
  • M. J. Alconada Verzini
  • M. Aleksa
  • I. N. Aleksandrov
  • C. Alexa
  • G. Alexander
  • T. Alexopoulos
  • M. Alhroob
  • B. Ali
  • M. Aliev
  • G. Alimonti
  • J. Alison
  • S. P. Alkire
  • B. M. M. Allbrooke
  • B. W. Allen
  • P. P. Allport
  • A. Aloisio
  • A. Alonso
  • F. Alonso
  • C. Alpigiani
  • A. A. Alshehri
  • M. Alstaty
  • B. Alvarez Gonzalez
  • D. Álvarez Piqueras
  • M. G. Alviggi
  • B. T. Amadio
  • Y. Amaral Coutinho
  • C. Amelung
  • D. Amidei
  • S. P. Amor Dos Santos
  • A. Amorim
  • S. Amoroso
  • G. Amundsen
  • C. Anastopoulos
  • L. S. Ancu
  • N. Andari
  • T. Andeen
  • C. F. Anders
  • J. K. Anders
  • K. J. Anderson
  • A. Andreazza
  • V. Andrei
  • S. Angelidakis
  • I. Angelozzi
  • A. Angerami
  • F. Anghinolfi
  • A. V. Anisenkov
  • N. Anjos
  • A. Annovi
  • C. Antel
  • M. Antonelli
  • A. Antonov
  • D. J. Antrim
  • F. Anulli
  • M. Aoki
  • L. Aperio Bella
  • G. Arabidze
  • Y. Arai
  • J. P. Araque
  • V. Araujo Ferraz
  • A. T. H. Arce
  • R. E. Ardell
  • F. A. Arduh
  • J-F. Arguin
  • S. Argyropoulos
  • M. Arik
  • A. J. Armbruster
  • L. J. Armitage
  • O. Arnaez
  • H. Arnold
  • M. Arratia
  • O. Arslan
  • A. Artamonov
  • G. Artoni
  • S. Artz
  • S. Asai
  • N. Asbah
  • A. Ashkenazi
  • L. Asquith
  • K. Assamagan
  • R. Astalos
  • M. Atkinson
  • N. B. Atlay
  • K. Augsten
  • G. Avolio
  • B. Axen
  • M. K. Ayoub
  • G. Azuelos
  • A. E. Baas
  • M. J. Baca
  • H. Bachacou
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  • M. Backhaus
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  • A. A. E. Bannoura
  • L. Barak
  • E. L. Barberio
  • D. Barberis
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  • B. M. Barnett
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  • A. Baroncelli
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  • L. Barranco Navarro
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  • J. Barreiro Guimarães da Costa
  • R. Bartoldus
  • A. E. Barton
  • P. Bartos
  • A. Basalaev
  • A. Bassalat
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  • E. Benhar Noccioli
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  • E. Bergeaas Kuutmann
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  • F. U. Bernlochner
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  • P. Berta
  • C. Bertella
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  • F. Bertolucci
  • I. A. Bertram
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  • D. Bertsche
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  • O. Bessidskaia Bylund
  • M. Bessner
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  • A. J. Bevan
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  • D. Biedermann
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  • N. V. Biesuz
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  • J. Bilbao De Mendizabal
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  • T. Bisanz
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  • D. M. Bjergaard
  • C. W. Black
  • J. E. Black
  • K. M. Black
  • D. Blackburn
  • R. E. Blair
  • J.-B. Blanchard
  • T. Blazek
  • I. Bloch
  • C. Blocker
  • A. Blue
  • W. Blum
  • U. Blumenschein
  • S. Blunier
  • G. J. Bobbink
  • V. S. Bobrovnikov
  • S. S. Bocchetta
  • A. Bocci
  • C. Bock
  • M. Boehler
  • D. Boerner
  • D. Bogavac
  • A. G. Bogdanchikov
  • C. Bohm
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  • A. S. Boldyrev
  • M. Bomben
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  • A. Borisov
  • G. Borissov
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  • D. Bortoletto
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  • I. R. Boyko
  • J. Bracinik
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  • B. Brau
  • J. E. Brau
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  • K. Brendlinger
  • A. J. Brennan
  • L. Brenner
  • R. Brenner
  • S. Bressler
  • D. L. Briglin
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  • D. Britton
  • D. Britzger
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  • I. Brock
  • R. Brock
  • G. Brooijmans
  • T. Brooks
  • W. K. Brooks
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  • E. Brost
  • J. H Broughton
  • P. A. Bruckman de Renstrom
  • D. Bruncko
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  • G. Bruni
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  • B H Brunt
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  • L. Bryngemark
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  • Q. Buat
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  • A. G. Buckley
  • I. A. Budagov
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  • M. K. Bugge
  • O. Bulekov
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  • J. T. P. Burr
  • E. Busato
  • D. Büscher
  • V. Büscher
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  • J. M. Butler
  • C. M. Buttar
  • J. M. Butterworth
  • P. Butti
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  • A. Buzatu
  • A. R. Buzykaev
  • S. Cabrera Urbán
  • D. Caforio
  • V. M. Cairo
  • O. Cakir
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  • P. Calafiura
  • A. Calandri
  • G. Calderini
  • P. Calfayan
  • G. Callea
  • L. P. Caloba
  • S. Calvente Lopez
  • D. Calvet
  • S. Calvet
  • T. P. Calvet
  • R. Camacho Toro
  • S. Camarda
  • P. Camarri
  • D. Cameron
  • R. Caminal Armadans
  • C. Camincher
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  • E. Cheremushkina
  • R. Cherkaoui El Moursli
  • V. Chernyatin
  • E. Cheu
  • L. Chevalier
  • V. Chiarella
  • G. Chiarelli
  • G. Chiodini
  • A. S. Chisholm
  • A. Chitan
  • Y. H. Chiu
  • M. V. Chizhov
  • K. Choi
  • A. R. Chomont
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  • B. K. B. Chow
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  • D. Chromek-Burckhart
  • M. C. Chu
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  • D. Cinca
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  • I. A. Cioara
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  • A. P. Colijn
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  • T. Colombo
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  • I. A. Connelly
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  • S. Constantinescu
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  • F. Conventi
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  • A. M. Cooper-Sarkar
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  • M. El Kacimi
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  • B. Esposito
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  • E. M. Farina
  • T. Farooque
  • S. Farrell
  • S. M. Farrington
  • P. Farthouat
  • F. Fassi
  • P. Fassnacht
  • D. Fassouliotis
  • M. Faucci Giannelli
  • A. Favareto
  • W. J. Fawcett
  • L. Fayard
  • O. L. Fedin
  • W. Fedorko
  • S. Feigl
  • L. Feligioni
  • C. Feng
  • E. J. Feng
  • H. Feng
  • A. B. Fenyuk
  • L. Feremenga
  • P. Fernandez Martinez
  • S. Fernandez Perez
  • J. Ferrando
  • A. Ferrari
  • P. Ferrari
  • R. Ferrari
  • D. E. Ferreira de Lima
  • A. Ferrer
  • D. Ferrere
  • C. Ferretti
  • F. Fiedler
  • A. Filipčič
  • M. Filipuzzi
  • F. Filthaut
  • M. Fincke-Keeler
  • K. D. Finelli
  • M. C. N. Fiolhais
  • L. Fiorini
  • A. Fischer
  • C. Fischer
  • J. Fischer
  • W. C. Fisher
  • N. Flaschel
  • I. Fleck
  • P. Fleischmann
  • R. R. M. Fletcher
  • T. Flick
  • B. M. Flierl
  • L. R. Flores Castillo
  • M. J. Flowerdew
  • G. T. Forcolin
  • A. Formica
  • A. Forti
  • A. G. Foster
  • D. Fournier
  • H. Fox
  • S. Fracchia
  • P. Francavilla
  • M. Franchini
  • D. Francis
  • L. Franconi
  • M. Franklin
  • M. Frate
  • M. Fraternali
  • D. Freeborn
  • S. M. Fressard-Batraneanu
  • B. Freund
  • D. Froidevaux
  • J. A. Frost
  • C. Fukunaga
  • E. Fullana Torregrosa
  • T. Fusayasu
  • J. Fuster
  • C. Gabaldon
  • O. Gabizon
  • A. Gabrielli
  • A. Gabrielli
  • G. P. Gach
  • S. Gadatsch
  • S. Gadomski
  • G. Gagliardi
  • L. G. Gagnon
  • P. Gagnon
  • C. Galea
  • B. Galhardo
  • E. J. Gallas
  • B. J. Gallop
  • P. Gallus
  • G. Galster
  • K. K. Gan
  • S. Ganguly
  • J. Gao
  • Y. Gao
  • Y. S. Gao
  • F. M. Garay Walls
  • C. García
  • J. E. García Navarro
  • M. Garcia-Sciveres
  • R. W. Gardner
  • N. Garelli
  • V. Garonne
  • A. Gascon Bravo
  • K. Gasnikova
  • C. Gatti
  • A. Gaudiello
  • G. Gaudio
  • I. L. Gavrilenko
  • C. Gay
  • G. Gaycken
  • E. N. Gazis
  • C. N. P. Gee
  • M. Geisen
  • M. P. Geisler
  • K. Gellerstedt
  • C. Gemme
  • M. H. Genest
  • C. Geng
  • S. Gentile
  • C. Gentsos
  • S. George
  • D. Gerbaudo
  • A. Gershon
  • S. Ghasemi
  • M. Ghneimat
  • B. Giacobbe
  • S. Giagu
  • P. Giannetti
  • S. M. Gibson
  • M. Gignac
  • M. Gilchriese
  • D. Gillberg
  • G. Gilles
  • D. M. Gingrich
  • N. Giokaris
  • M. P. Giordani
  • F. M. Giorgi
  • P. F. Giraud
  • P. Giromini
  • D. Giugni
  • F. Giuli
  • C. Giuliani
  • M. Giulini
  • B. K. Gjelsten
  • S. Gkaitatzis
  • I. Gkialas
  • E. L. Gkougkousis
  • L. K. Gladilin
  • C. Glasman
  • J. Glatzer
  • P. C. F. Glaysher
  • A. Glazov
  • M. Goblirsch-Kolb
  • J. Godlewski
  • S. Goldfarb
  • T. Golling
  • D. Golubkov
  • A. Gomes
  • R. Gonçalo
  • R. Goncalves Gama
  • J. Goncalves Pinto Firmino Da Costa
  • G. Gonella
  • L. Gonella
  • A. Gongadze
  • S. González de la Hoz
  • S. Gonzalez-Sevilla
  • L. Goossens
  • P. A. Gorbounov
  • H. A. Gordon
  • I. Gorelov
  • B. Gorini
  • E. Gorini
  • A. Gorišek
  • A. T. Goshaw
  • C. Gössling
  • M. I. Gostkin
  • C. R. Goudet
  • D. Goujdami
  • A. G. Goussiou
  • N. Govender
  • E. Gozani
  • L. Graber
  • I. Grabowska-Bold
  • P. O. J. Gradin
  • J. Gramling
  • E. Gramstad
  • S. Grancagnolo
  • V. Gratchev
  • P. M. Gravila
  • H. M. Gray
  • Z. D. Greenwood
  • C. Grefe
  • K. Gregersen
  • I. M. Gregor
  • P. Grenier
  • K. Grevtsov
  • J. Griffiths
  • A. A. Grillo
  • K. Grimm
  • S. Grinstein
  • Ph. Gris
  • J.-F. Grivaz
  • S. Groh
  • E. Gross
  • J. Grosse-Knetter
  • G. C. Grossi
  • Z. J. Grout
  • L. Guan
  • W. Guan
  • J. Guenther
  • F. Guescini
  • D. Guest
  • O. Gueta
  • B. Gui
  • E. Guido
  • T. Guillemin
  • S. Guindon
  • U. Gul
  • C. Gumpert
  • J. Guo
  • W. Guo
  • Y. Guo
  • R. Gupta
  • S. Gupta
  • G. Gustavino
  • P. Gutierrez
  • N. G. Gutierrez Ortiz
  • C. Gutschow
  • C. Guyot
  • M. P. Guzik
  • C. Gwenlan
  • C. B. Gwilliam
  • A. Haas
  • C. Haber
  • H. K. Hadavand
  • A. Hadef
  • S. Hageböck
  • M. Hagihara
  • H. Hakobyan
  • M. Haleem
  • J. Haley
  • G. Halladjian
  • G. D. Hallewell
  • K. Hamacher
  • P. Hamal
  • K. Hamano
  • A. Hamilton
  • G. N. Hamity
  • P. G. Hamnett
  • L. Han
  • S. Han
  • K. Hanagaki
  • K. Hanawa
  • M. Hance
  • B. Haney
  • P. Hanke
  • R. Hanna
  • J. B. Hansen
  • J. D. Hansen
  • M. C. Hansen
  • P. H. Hansen
  • K. Hara
  • A. S. Hard
  • T. Harenberg
  • F. Hariri
  • S. Harkusha
  • R. D. Harrington
  • P. F. Harrison
  • F. Hartjes
  • N. M. Hartmann
  • M. Hasegawa
  • Y. Hasegawa
  • A. Hasib
  • S. Hassani
  • S. Haug
  • R. Hauser
  • L. Hauswald
  • L. B. Havener
  • M. Havranek
  • C. M. Hawkes
  • R. J. Hawkings
  • D. Hayakawa
  • D. Hayden
  • C. P. Hays
  • J. M. Hays
  • H. S. Hayward
  • S. J. Haywood
  • S. J. Head
  • T. Heck
  • V. Hedberg
  • L. Heelan
  • K. K. Heidegger
  • S. Heim
  • T. Heim
  • B. Heinemann
  • J. J. Heinrich
  • L. Heinrich
  • C. Heinz
  • J. Hejbal
  • L. Helary
  • A. Held
  • S. Hellman
  • C. Helsens
  • J. Henderson
  • R. C. W. Henderson
  • Y. Heng
  • S. Henkelmann
  • A. M. Henriques Correia
  • S. Henrot-Versille
  • G. H. Herbert
  • H. Herde
  • V. Herget
  • Y. Hernández Jiménez
  • G. Herten
  • R. Hertenberger
  • L. Hervas
  • T. C. Herwig
  • G. G. Hesketh
  • N. P. Hessey
  • J. W. Hetherly
  • S. Higashino
  • E. Higón-Rodriguez
  • E. Hill
  • J. C. Hill
  • K. H. Hiller
  • S. J. Hillier
  • I. Hinchliffe
  • M. Hirose
  • D. Hirschbuehl
  • B. Hiti
  • O. Hladik
  • X. Hoad
  • J. Hobbs
  • N. Hod
  • M. C. Hodgkinson
  • P. Hodgson
  • A. Hoecker
  • M. R. Hoeferkamp
  • F. Hoenig
  • D. Hohn
  • T. R. Holmes
  • M. Homann
  • S. Honda
  • T. Honda
  • T. M. Hong
  • B. H. Hooberman
  • W. H. Hopkins
  • Y. Horii
  • A. J. Horton
  • J-Y. Hostachy
  • S. Hou
  • A. Hoummada
  • J. Howarth
  • J. Hoya
  • M. Hrabovsky
  • I. Hristova
  • J. Hrivnac
  • T. Hryn’ova
  • A. Hrynevich
  • P. J. Hsu
  • S.-C. Hsu
  • Q. Hu
  • S. Hu
  • Y. Huang
  • Z. Hubacek
  • F. Hubaut
  • F. Huegging
  • T. B. Huffman
  • E. W. Hughes
  • G. Hughes
  • M. Huhtinen
  • P. Huo
  • N. Huseynov
  • J. Huston
  • J. Huth
  • G. Iacobucci
  • G. Iakovidis
  • I. Ibragimov
  • L. Iconomidou-Fayard
  • P. Iengo
  • O. Igonkina
  • T. Iizawa
  • Y. Ikegami
  • M. Ikeno
  • Y. Ilchenko
  • D. Iliadis
  • N. Ilic
  • G. Introzzi
  • P. Ioannou
  • M. Iodice
  • K. Iordanidou
  • V. Ippolito
  • N. Ishijima
  • M. Ishino
  • M. Ishitsuka
  • C. Issever
  • S. Istin
  • F. Ito
  • J. M. Iturbe Ponce
  • R. Iuppa
  • H. Iwasaki
  • J. M. Izen
  • V. Izzo
  • S. Jabbar
  • P. Jackson
  • V. Jain
  • K. B. Jakobi
  • K. Jakobs
  • S. Jakobsen
  • T. Jakoubek
  • D. O. Jamin
  • D. K. Jana
  • R. Jansky
  • J. Janssen
  • M. Janus
  • P. A. Janus
  • G. Jarlskog
  • N. Javadov
  • T. Javůrek
  • M. Javurkova
  • F. Jeanneau
  • L. Jeanty
  • J. Jejelava
  • A. Jelinskas
  • P. Jenni
  • C. Jeske
  • S. Jézéquel
  • H. Ji
  • J. Jia
  • H. Jiang
  • Y. Jiang
  • Z. Jiang
  • S. Jiggins
  • J. Jimenez Pena
  • S. Jin
  • A. Jinaru
  • O. Jinnouchi
  • H. Jivan
  • P. Johansson
  • K. A. Johns
  • C. A. Johnson
  • W. J. Johnson
  • K. Jon-And
  • R. W. L. Jones
  • S. Jones
  • T. J. Jones
  • J. Jongmanns
  • P. M. Jorge
  • J. Jovicevic
  • X. Ju
  • A. Juste Rozas
  • M. K. Köhler
  • A. Kaczmarska
  • M. Kado
  • H. Kagan
  • M. Kagan
  • S. J. Kahn
  • T. Kaji
  • E. Kajomovitz
  • C. W. Kalderon
  • A. Kaluza
  • S. Kama
  • A. Kamenshchikov
  • N. Kanaya
  • S. Kaneti
  • L. Kanjir
  • V. A. Kantserov
  • J. Kanzaki
  • B. Kaplan
  • L. S. Kaplan
  • D. Kar
  • K. Karakostas
  • N. Karastathis
  • M. J. Kareem
  • E. Karentzos
  • M. Karnevskiy
  • S. N. Karpov
  • Z. M. Karpova
  • K. Karthik
  • V. Kartvelishvili
  • A. N. Karyukhin
  • K. Kasahara
  • L. Kashif
  • R. D. Kass
  • A. Kastanas
  • Y. Kataoka
  • C. Kato
  • A. Katre
  • J. Katzy
  • K. Kawade
  • K. Kawagoe
  • T. Kawamoto
  • G. Kawamura
  • E. F. Kay
  • V. F. Kazanin
  • R. Keeler
  • R. Kehoe
  • J. S. Keller
  • J. J. Kempster
  • H. Keoshkerian
  • O. Kepka
  • B. P. Kerševan
  • S. Kersten
  • R. A. Keyes
  • M. Khader
  • F. Khalil-zada
  • A. Khanov
  • A. G. Kharlamov
  • T. Kharlamova
  • A. Khodinov
  • T. J. Khoo
  • V. Khovanskiy
  • E. Khramov
  • J. Khubua
  • S. Kido
  • C. R. Kilby
  • H. Y. Kim
  • S. H. Kim
  • Y. K. Kim
  • N. Kimura
  • O. M. Kind
  • B. T. King
  • D. Kirchmeier
  • J. Kirk
  • A. E. Kiryunin
  • T. Kishimoto
  • D. Kisielewska
  • K. Kiuchi
  • O. Kivernyk
  • E. Kladiva
  • T. Klapdor-Kleingrothaus
  • M. H. Klein
  • M. Klein
  • U. Klein
  • K. Kleinknecht
  • P. Klimek
  • A. Klimentov
  • R. Klingenberg
  • T. Klioutchnikova
  • E.-E. Kluge
  • P. Kluit
  • S. Kluth
  • J. Knapik
  • E. Kneringer
  • E. B. F. G. Knoops
  • A. Knue
  • A. Kobayashi
  • D. Kobayashi
  • T. Kobayashi
  • M. Kobel
  • M. Kocian
  • P. Kodys
  • T. Koffas
  • E. Koffeman
  • N. M. Köhler
  • T. Koi
  • M. Kolb
  • I. Koletsou
  • A. A. Komar
  • Y. Komori
  • T. Kondo
  • N. Kondrashova
  • K. Köneke
  • A. C. König
  • T. Kono
  • R. Konoplich
  • N. Konstantinidis
  • R. Kopeliansky
  • S. Koperny
  • A. K. Kopp
  • K. Korcyl
  • K. Kordas
  • A. Korn
  • A. A. Korol
  • I. Korolkov
  • E. V. Korolkova
  • O. Kortner
  • S. Kortner
  • T. Kosek
  • V. V. Kostyukhin
  • A. Kotwal
  • A. Koulouris
  • A. Kourkoumeli-Charalampidi
  • C. Kourkoumelis
  • V. Kouskoura
  • A. B. Kowalewska
  • R. Kowalewski
  • T. Z. Kowalski
  • C. Kozakai
  • W. Kozanecki
  • A. S. Kozhin
  • V. A. Kramarenko
  • G. Kramberger
  • D. Krasnopevtsev
  • M. W. Krasny
  • A. Krasznahorkay
  • D. Krauss
  • A. Kravchenko
  • J. A. Kremer
  • M. Kretz
  • J. Kretzschmar
  • K. Kreutzfeldt
  • P. Krieger
  • K. Krizka
  • K. Kroeninger
  • H. Kroha
  • J. Kroll
  • J. Kroseberg
  • J. Krstic
  • U. Kruchonak
  • H. Krüger
  • N. Krumnack
  • M. C. Kruse
  • M. Kruskal
  • T. Kubota
  • H. Kucuk
  • S. Kuday
  • J. T. Kuechler
  • S. Kuehn
  • A. Kugel
  • F. Kuger
  • T. Kuhl
  • V. Kukhtin
  • R. Kukla
  • Y. Kulchitsky
  • S. Kuleshov
  • Y. P. Kulinich
  • M. Kuna
  • T. Kunigo
  • A. Kupco
  • O. Kuprash
  • H. Kurashige
  • L. L. Kurchaninov
  • Y. A. Kurochkin
  • M. G. Kurth
  • V. Kus
  • E. S. Kuwertz
  • M. Kuze
  • J. Kvita
  • T. Kwan
  • D. Kyriazopoulos
  • A. La Rosa
  • J. L. La Rosa Navarro
  • L. La Rotonda
  • C. Lacasta
  • F. Lacava
  • J. Lacey
  • H. Lacker
  • D. Lacour
  • E. Ladygin
  • R. Lafaye
  • B. Laforge
  • T. Lagouri
  • S. Lai
  • S. Lammers
  • W. Lampl
  • E. Lançon
  • U. Landgraf
  • M. P. J. Landon
  • M. C. Lanfermann
  • V. S. Lang
  • J. C. Lange
  • A. J. Lankford
  • F. Lanni
  • K. Lantzsch
  • A. Lanza
  • A. Lapertosa
  • S. Laplace
  • J. F. Laporte
  • T. Lari
  • F. Lasagni Manghi
  • M. Lassnig
  • P. Laurelli
  • W. Lavrijsen
  • A. T. Law
  • P. Laycock
  • T. Lazovich
  • M. Lazzaroni
  • B. Le
  • O. Le Dortz
  • E. Le Guirriec
  • E. P. Le Quilleuc
  • M. LeBlanc
  • T. LeCompte
  • F. Ledroit-Guillon
  • C. A. Lee
  • S. C. Lee
  • L. Lee
  • B. Lefebvre
  • G. Lefebvre
  • M. Lefebvre
  • F. Legger
  • C. Leggett
  • A. Lehan
  • G. Lehmann Miotto
  • X. Lei
  • W. A. Leight
  • A. G. Leister
  • M. A. L. Leite
  • R. Leitner
  • D. Lellouch
  • B. Lemmer
  • K. J. C. Leney
  • T. Lenz
  • B. Lenzi
  • R. Leone
  • S. Leone
  • C. Leonidopoulos
  • G. Lerner
  • C. Leroy
  • A. A. J. Lesage
  • C. G. Lester
  • M. Levchenko
  • J. Levêque
  • D. Levin
  • L. J. Levinson
  • M. Levy
  • D. Lewis
  • M. Leyton
  • B. Li
  • C. Li
  • H. Li
  • L. Li
  • L. Li
  • Q. Li
  • S. Li
  • X. Li
  • Y. Li
  • Z. Liang
  • B. Liberti
  • A. Liblong
  • K. Lie
  • J. Liebal
  • W. Liebig
  • A. Limosani
  • S. C. Lin
  • T. H. Lin
  • B. E. Lindquist
  • A. E. Lionti
  • E. Lipeles
  • A. Lipniacka
  • M. Lisovyi
  • T. M. Liss
  • A. Lister
  • A. M. Litke
  • B. Liu
  • H. Liu
  • H. Liu
  • J. Liu
  • J. B. Liu
  • K. Liu
  • L. Liu
  • M. Liu
  • Y. L. Liu
  • Y. Liu
  • M. Livan
  • A. Lleres
  • J. Llorente Merino
  • S. L. Lloyd
  • C. Y. Lo
  • F. Lo Sterzo
  • E. M. Lobodzinska
  • P. Loch
  • F. K. Loebinger
  • K. M. Loew
  • A. Loginov
  • T. Lohse
  • K. Lohwasser
  • M. Lokajicek
  • B. A. Long
  • J. D. Long
  • R. E. Long
  • L. Longo
  • K. A. Looper
  • J. A. Lopez
  • D. Lopez Mateos
  • I. Lopez Paz
  • A. Lopez Solis
  • J. Lorenz
  • N. Lorenzo Martinez
  • M. Losada
  • P. J. Lösel
  • X. Lou
  • A. Lounis
  • J. Love
  • P. A. Love
  • H. Lu
  • N. Lu
  • Y. J. Lu
  • H. J. Lubatti
  • C. Luci
  • A. Lucotte
  • C. Luedtke
  • F. Luehring
  • W. Lukas
  • L. Luminari
  • O. Lundberg
  • B. Lund-Jensen
  • P. M. Luzi
  • D. Lynn
  • R. Lysak
  • E. Lytken
  • V. Lyubushkin
  • H. Ma
  • L. L. Ma
  • Y. Ma
  • G. Maccarrone
  • A. Macchiolo
  • C. M. Macdonald
  • B. Maček
  • J. Machado Miguens
  • D. Madaffari
  • R. Madar
  • H. J. Maddocks
  • W. F. Mader
  • A. Madsen
  • J. Maeda
  • S. Maeland
  • T. Maeno
  • A. Maevskiy
  • E. Magradze
  • J. Mahlstedt
  • C. Maiani
  • C. Maidantchik
  • A. A. Maier
  • T. Maier
  • A. Maio
  • S. Majewski
  • Y. Makida
  • N. Makovec
  • B. Malaescu
  • Pa. Malecki
  • V. P. Maleev
  • F. Malek
  • U. Mallik
  • D. Malon
  • C. Malone
  • S. Maltezos
  • S. Malyukov
  • J. Mamuzic
  • G. Mancini
  • L. Mandelli
  • I. Mandić
  • J. Maneira
  • L. Manhaes de Andrade Filho
  • J. Manjarres Ramos
  • A. Mann
  • A. Manousos
  • B. Mansoulie
  • J. D. Mansour
  • R. Mantifel
  • M. Mantoani
  • S. Manzoni
  • L. Mapelli
  • G. Marceca
  • L. March
  • G. Marchiori
  • M. Marcisovsky
  • M. Marjanovic
  • D. E. Marley
  • F. Marroquim
  • S. P. Marsden
  • Z. Marshall
  • M. U. F Martensson
  • S. Marti-Garcia
  • C. B. Martin
  • T. A. Martin
  • V. J. Martin
  • B. Martin dit Latour
  • M. Martinez
  • V. I. Martinez Outschoorn
  • S. Martin-Haugh
  • V. S. Martoiu
  • A. C. Martyniuk
  • A. Marzin
  • L. Masetti
  • T. Mashimo
  • R. Mashinistov
  • J. Masik
  • A. L. Maslennikov
  • L. Massa
  • P. Mastrandrea
  • A. Mastroberardino
  • T. Masubuchi
  • P. Mättig
  • J. Maurer
  • S. J. Maxfield
  • D. A. Maximov
  • R. Mazini
  • I. Maznas
  • S. M. Mazza
  • N. C. Mc Fadden
  • G. Mc Goldrick
  • S. P. Mc Kee
  • A. McCarn
  • R. L. McCarthy
  • T. G. McCarthy
  • L. I. McClymont
  • E. F. McDonald
  • J. A. Mcfayden
  • G. Mchedlidze
  • S. J. McMahon
  • P. C. McNamara
  • R. A. McPherson
  • S. Meehan
  • T. J. Megy
  • S. Mehlhase
  • A. Mehta
  • T. Meideck
  • K. Meier
  • C. Meineck
  • B. Meirose
  • D. Melini
  • B. R. Mellado Garcia
  • M. Melo
  • F. Meloni
  • S. B. Menary
  • L. Meng
  • X. T. Meng
  • A. Mengarelli
  • S. Menke
  • E. Meoni
  • S. Mergelmeyer
  • P. Mermod
  • L. Merola
  • C. Meroni
  • F. S. Merritt
  • A. Messina
  • J. Metcalfe
  • A. S. Mete
  • C. Meyer
  • J-P. Meyer
  • J. Meyer
  • H. Meyer Zu Theenhausen
  • F. Miano
  • R. P. Middleton
  • S. Miglioranzi
  • L. Mijović
  • G. Mikenberg
  • M. Mikestikova
  • M. Mikuž
  • M. Milesi
  • A. Milic
  • D. W. Miller
  • C. Mills
  • A. Milov
  • D. A. Milstead
  • A. A. Minaenko
  • Y. Minami
  • I. A. Minashvili
  • A. I. Mincer
  • B. Mindur
  • M. Mineev
  • Y. Minegishi
  • Y. Ming
  • L. M. Mir
  • K. P. Mistry
  • T. Mitani
  • J. Mitrevski
  • V. A. Mitsou
  • A. Miucci
  • P. S. Miyagawa
  • A. Mizukami
  • J. U. Mjörnmark
  • M. Mlynarikova
  • T. Moa
  • K. Mochizuki
  • P. Mogg
  • S. Mohapatra
  • S. Molander
  • R. Moles-Valls
  • R. Monden
  • M. C. Mondragon
  • K. Mönig
  • J. Monk
  • E. Monnier
  • A. Montalbano
  • J. Montejo Berlingen
  • F. Monticelli
  • S. Monzani
  • R. W. Moore
  • N. Morange
  • D. Moreno
  • M. Moreno Llácer
  • P. Morettini
  • S. Morgenstern
  • D. Mori
  • T. Mori
  • M. Morii
  • M. Morinaga
  • V. Morisbak
  • A. K. Morley
  • G. Mornacchi
  • J. D. Morris
  • L. Morvaj
  • P. Moschovakos
  • M. Mosidze
  • H. J. Moss
  • J. Moss
  • K. Motohashi
  • R. Mount
  • E. Mountricha
  • E. J. W. Moyse
  • S. Muanza
  • R. D. Mudd
  • F. Mueller
  • J. Mueller
  • R. S. P. Mueller
  • D. Muenstermann
  • P. Mullen
  • G. A. Mullier
  • F. J. Munoz Sanchez
  • W. J. Murray
  • H. Musheghyan
  • M. Muškinja
  • A. G. Myagkov
  • M. Myska
  • B. P. Nachman
  • O. Nackenhorst
  • K. Nagai
  • R. Nagai
  • K. Nagano
  • Y. Nagasaka
  • K. Nagata
  • M. Nagel
  • E. Nagy
  • A. M. Nairz
  • Y. Nakahama
  • K. Nakamura
  • T. Nakamura
  • I. Nakano
  • R. F. Naranjo Garcia
  • R. Narayan
  • D. I. Narrias Villar
  • I. Naryshkin
  • T. Naumann
  • G. Navarro
  • R. Nayyar
  • H. A. Neal
  • P. Yu. Nechaeva
  • T. J. Neep
  • A. Negri
  • M. Negrini
  • S. Nektarijevic
  • C. Nellist
  • A. Nelson
  • M. E. Nelson
  • S. Nemecek
  • P. Nemethy
  • A. A. Nepomuceno
  • M. Nessi
  • M. S. Neubauer
  • M. Neumann
  • R. M. Neves
  • P. Nevski
  • P. R. Newman
  • T. Y. Ng
  • T. Nguyen Manh
  • R. B. Nickerson
  • R. Nicolaidou
  • J. Nielsen
  • V. Nikolaenko
  • I. Nikolic-Audit
  • K. Nikolopoulos
  • J. K. Nilsen
  • P. Nilsson
  • Y. Ninomiya
  • A. Nisati
  • N. Nishu
  • R. Nisius
  • T. Nobe
  • Y. Noguchi
  • M. Nomachi
  • I. Nomidis
  • M. A. Nomura
  • T. Nooney
  • M. Nordberg
  • N. Norjoharuddeen
  • O. Novgorodova
  • S. Nowak
  • M. Nozaki
  • L. Nozka
  • K. Ntekas
  • E. Nurse
  • F. Nuti
  • D. C. O’Neil
  • A. A. O’Rourke
  • V. O’Shea
  • F. G. Oakham
  • H. Oberlack
  • T. Obermann
  • J. Ocariz
  • A. Ochi
  • I. Ochoa
  • J. P. Ochoa-Ricoux
  • S. Oda
  • S. Odaka
  • H. Ogren
  • A. Oh
  • S. H. Oh
  • C. C. Ohm
  • H. Ohman
  • H. Oide
  • H. Okawa
  • Y. Okumura
  • T. Okuyama
  • A. Olariu
  • L. F. Oleiro Seabra
  • S. A. Olivares Pino
  • D. Oliveira Damazio
  • A. Olszewski
  • J. Olszowska
  • A. Onofre
  • K. Onogi
  • P. U. E. Onyisi
  • M. J. Oreglia
  • Y. Oren
  • D. Orestano
  • N. Orlando
  • R. S. Orr
  • B. Osculati
  • R. Ospanov
  • G. Otero y Garzon
  • H. Otono
  • M. Ouchrif
  • F. Ould-Saada
  • A. Ouraou
  • K. P. Oussoren
  • Q. Ouyang
  • M. Owen
  • R. E. Owen
  • V. E. Ozcan
  • N. Ozturk
  • K. Pachal
  • A. Pacheco Pages
  • L. Pacheco Rodriguez
  • C. Padilla Aranda
  • S. Pagan Griso
  • M. Paganini
  • F. Paige
  • P. Pais
  • G. Palacino
  • S. Palazzo
  • S. Palestini
  • M. Palka
  • D. Pallin
  • E. St. Panagiotopoulou
  • I. Panagoulias
  • C. E. Pandini
  • J. G. Panduro Vazquez
  • P. Pani
  • S. Panitkin
  • D. Pantea
  • L. Paolozzi
  • Th. D. Papadopoulou
  • K. Papageorgiou
  • A. Paramonov
  • D. Paredes Hernandez
  • A. J. Parker
  • M. A. Parker
  • K. A. Parker
  • F. Parodi
  • J. A. Parsons
  • U. Parzefall
  • V. R. Pascuzzi
  • J. M. Pasner
  • E. Pasqualucci
  • S. Passaggio
  • Fr. Pastore
  • S. Pataraia
  • J. R. Pater
  • T. Pauly
  • J. Pearce
  • B. Pearson
  • L. E. Pedersen
  • S. Pedraza Lopez
  • R. Pedro
  • S. V. Peleganchuk
  • O. Penc
  • C. Peng
  • H. Peng
  • J. Penwell
  • B. S. Peralva
  • M. M. Perego
  • D. V. Perepelitsa
  • L. Perini
  • H. Pernegger
  • S. Perrella
  • R. Peschke
  • V. D. Peshekhonov
  • K. Peters
  • R. F. Y. Peters
  • B. A. Petersen
  • T. C. Petersen
  • E. Petit
  • A. Petridis
  • C. Petridou
  • P. Petroff
  • E. Petrolo
  • M. Petrov
  • F. Petrucci
  • N. E. Pettersson
  • A. Peyaud
  • R. Pezoa
  • P. W. Phillips
  • G. Piacquadio
  • E. Pianori
  • A. Picazio
  • E. Piccaro
  • M. A. Pickering
  • R. Piegaia
  • J. E. Pilcher
  • A. D. Pilkington
  • A. W. J. Pin
  • M. Pinamonti
  • J. L. Pinfold
  • H. Pirumov
  • M. Pitt
  • L. Plazak
  • M.-A. Pleier
  • V. Pleskot
  • E. Plotnikova
  • D. Pluth
  • P. Podberezko
  • R. Poettgen
  • L. Poggioli
  • D. Pohl
  • G. Polesello
  • A. Poley
  • A. Policicchio
  • R. Polifka
  • A. Polini
  • C. S. Pollard
  • V. Polychronakos
  • K. Pommès
  • L. Pontecorvo
  • B. G. Pope
  • G. A. Popeneciu
  • A. Poppleton
  • S. Pospisil
  • K. Potamianos
  • I. N. Potrap
  • C. J. Potter
  • C. T. Potter
  • G. Poulard
  • J. Poveda
  • M. E. Pozo Astigarraga
  • P. Pralavorio
  • A. Pranko
  • S. Prell
  • D. Price
  • L. E. Price
  • M. Primavera
  • S. Prince
  • K. Prokofiev
  • F. Prokoshin
  • S. Protopopescu
  • J. Proudfoot
  • M. Przybycien
  • D. Puddu
  • A. Puri
  • P. Puzo
  • J. Qian
  • G. Qin
  • Y. Qin
  • A. Quadt
  • W. B. Quayle
  • M. Queitsch-Maitland
  • D. Quilty
  • S. Raddum
  • V. Radeka
  • V. Radescu
  • S. K. Radhakrishnan
  • P. Radloff
  • P. Rados
  • F. Ragusa
  • G. Rahal
  • J. A. Raine
  • S. Rajagopalan
  • C. Rangel-Smith
  • M. G. Ratti
  • D. M. Rauch
  • F. Rauscher
  • S. Rave
  • T. Ravenscroft
  • I. Ravinovich
  • M. Raymond
  • A. L. Read
  • N. P. Readioff
  • M. Reale
  • D. M. Rebuzzi
  • A. Redelbach
  • G. Redlinger
  • R. Reece
  • R. G. Reed
  • K. Reeves
  • L. Rehnisch
  • J. Reichert
  • A. Reiss
  • C. Rembser
  • H. Ren
  • M. Rescigno
  • S. Resconi
  • E. D. Resseguie
  • S. Rettie
  • E. Reynolds
  • O. L. Rezanova
  • P. Reznicek
  • R. Rezvani
  • R. Richter
  • S. Richter
  • E. Richter-Was
  • O. Ricken
  • M. Ridel
  • P. Rieck
  • C. J. Riegel
  • J. Rieger
  • O. Rifki
  • M. Rijssenbeek
  • A. Rimoldi
  • M. Rimoldi
  • L. Rinaldi
  • B. Ristić
  • E. Ritsch
  • I. Riu
  • F. Rizatdinova
  • E. Rizvi
  • C. Rizzi
  • R. T. Roberts
  • S. H. Robertson
  • A. Robichaud-Veronneau
  • D. Robinson
  • J. E. M. Robinson
  • A. Robson
  • C. Roda
  • Y. Rodina
  • A. Rodriguez Perez
  • D. Rodriguez Rodriguez
  • S. Roe
  • C. S. Rogan
  • O. Røhne
  • J. Roloff
  • A. Romaniouk
  • M. Romano
  • S. M. Romano Saez
  • E. Romero Adam
  • N. Rompotis
  • M. Ronzani
  • L. Roos
  • S. Rosati
  • K. Rosbach
  • P. Rose
  • N.-A. Rosien
  • V. Rossetti
  • E. Rossi
  • L. P. Rossi
  • J. H. N. Rosten
  • R. Rosten
  • M. Rotaru
  • I. Roth
  • J. Rothberg
  • D. Rousseau
  • A. Rozanov
  • Y. Rozen
  • X. Ruan
  • F. Rubbo
  • F. Rühr
  • A. Ruiz-Martinez
  • Z. Rurikova
  • N. A. Rusakovich
  • A. Ruschke
  • H. L. Russell
  • J. P. Rutherfoord
  • N. Ruthmann
  • Y. F. Ryabov
  • M. Rybar
  • G. Rybkin
  • S. Ryu
  • A. Ryzhov
  • G. F. Rzehorz
  • A. F. Saavedra
  • G. Sabato
  • S. Sacerdoti
  • H. F-W. Sadrozinski
  • R. Sadykov
  • F. Safai Tehrani
  • P. Saha
  • M. Sahinsoy
  • M. Saimpert
  • M. Saito
  • T. Saito
  • H. Sakamoto
  • Y. Sakurai
  • G. Salamanna
  • J. E. Salazar Loyola
  • D. Salek
  • P. H. Sales De Bruin
  • D. Salihagic
  • A. Salnikov
  • J. Salt
  • D. Salvatore
  • F. Salvatore
  • A. Salvucci
  • A. Salzburger
  • D. Sammel
  • D. Sampsonidis
  • J. Sánchez
  • V. Sanchez Martinez
  • A. Sanchez Pineda
  • H. Sandaker
  • R. L. Sandbach
  • C. O. Sander
  • M. Sandhoff
  • C. Sandoval
  • D. P. C. Sankey
  • M. Sannino
  • A. Sansoni
  • C. Santoni
  • R. Santonico
  • H. Santos
  • I. Santoyo Castillo
  • K. Sapp
  • A. Sapronov
  • J. G. Saraiva
  • B. Sarrazin
  • O. Sasaki
  • K. Sato
  • E. Sauvan
  • G. Savage
  • P. Savard
  • N. Savic
  • C. Sawyer
  • L. Sawyer
  • J. Saxon
  • C. Sbarra
  • A. Sbrizzi
  • T. Scanlon
  • D. A. Scannicchio
  • M. Scarcella
  • V. Scarfone
  • J. Schaarschmidt
  • P. Schacht
  • B. M. Schachtner
  • D. Schaefer
  • L. Schaefer
  • R. Schaefer
  • J. Schaeffer
  • S. Schaepe
  • S. Schaetzel
  • U. Schäfer
  • A. C. Schaffer
  • D. Schaile
  • R. D. Schamberger
  • V. Scharf
  • V. A. Schegelsky
  • D. Scheirich
  • M. Schernau
  • C. Schiavi
  • S. Schier
  • C. Schillo
  • M. Schioppa
  • S. Schlenker
  • K. R. Schmidt-Sommerfeld
  • K. Schmieden
  • C. Schmitt
  • S. Schmitt
  • S. Schmitz
  • B. Schneider
  • U. Schnoor
  • L. Schoeffel
  • A. Schoening
  • B. D. Schoenrock
  • E. Schopf
  • M. Schott
  • J. F. P. Schouwenberg
  • J. Schovancova
  • S. Schramm
  • N. Schuh
  • A. Schulte
  • M. J. Schultens
  • H.-C. Schultz-Coulon
  • H. Schulz
  • M. Schumacher
  • B. A. Schumm
  • Ph. Schune
  • A. Schwartzman
  • T. A. Schwarz
  • H. Schweiger
  • Ph. Schwemling
  • R. Schwienhorst
  • J. Schwindling
  • T. Schwindt
  • G. Sciolla
  • F. Scuri
  • F. Scutti
  • J. Searcy
  • P. Seema
  • S. C. Seidel
  • A. Seiden
  • J. M. Seixas
  • G. Sekhniaidze
  • K. Sekhon
  • S. J. Sekula
  • N. Semprini-Cesari
  • C. Serfon
  • L. Serin
  • L. Serkin
  • M. Sessa
  • R. Seuster
  • H. Severini
  • T. Sfiligoj
  • F. Sforza
  • A. Sfyrla
  • E. Shabalina
  • N. W. Shaikh
  • L. Y. Shan
  • R. Shang
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  • ATLAS Collaboration
Open Access
Regular Article - Experimental Physics

Abstract

A measurement of the mass of the W boson is presented based on proton–proton collision data recorded in 2011 at a centre-of-mass energy of 7 TeV with the ATLAS detector at the LHC, and corresponding to \(4.6~\hbox {fb}^{-1}\) of integrated luminosity. The selected data sample consists of \(7.8\times 10^6\) candidates in the \(W\rightarrow \mu \nu \) channel and \(5.9\times 10^6\) candidates in the \(W\rightarrow e \nu \) channel. The W-boson mass is obtained from template fits to the reconstructed distributions of the charged lepton transverse momentum and of the W boson transverse mass in the electron and muon decay channels, yielding
$$\begin{aligned} \nonumber m_W = 80370&\pm 7~(\text {stat.}) \pm 11 (\text {exp. syst.}) \\&\pm 14~(\text {mod. syst.})~\text {MeV} \\ = 80370&\pm 19\,\text {MeV}, \end{aligned}$$
where the first uncertainty is statistical, the second corresponds to the experimental systematic uncertainty, and the third to the physics-modelling systematic uncertainty. A measurement of the mass difference between the \(W^+\) and \(W^-\) bosons yields \(m_{W^+}-m_{W^-} = -\,29 \pm 28\) MeV.

1 Introduction

The Standard Model (SM) of particle physics describes the electroweak interactions as being mediated by the W boson, the Z boson, and the photon, in a gauge theory based on the \({\mathrm {SU}}(2)_{\mathrm {L}} \times {\mathrm {U}}(1)_{\mathrm {Y}}\) symmetry [1, 2, 3]. The theory incorporates the observed masses of the W and Z bosons through a symmetry-breaking mechanism. In the SM, this mechanism relies on the interaction of the gauge bosons with a scalar doublet field and implies the existence of an additional physical state known as the Higgs boson [4, 5, 6, 7]. The existence of the W and Z bosons was first established at the CERN SPS in 1983 [8, 9, 10, 11], and the LHC collaborations ATLAS and CMS reported the discovery of the Higgs boson in 2012 [12, 13].

At lowest order in the electroweak theory, the W-boson mass, \(m_W\), can be expressed solely as a function of the Z-boson mass, \(m_Z\), the fine-structure constant, \(\alpha \), and the Fermi constant, \(G_{\mu }\). Higher-order corrections introduce an additional dependence of the W-boson mass on the gauge couplings and the masses of the heavy particles of the SM. The mass of the W boson can be expressed in terms of the other SM parameters as follows:
$$\begin{aligned} \nonumber m_W^2 \left( 1 - \frac{m^2_W}{m^2_Z}\right) = \frac{\pi \alpha }{\sqrt{2}G_{\mu }} (1+\Delta r), \end{aligned}$$
where \(\Delta r\) incorporates the effect of higher-order corrections [14, 15]. In the SM, \(\Delta r\) is in particular sensitive to the top-quark and Higgs-boson masses; in extended theories, \(\Delta r\) receives contributions from additional particles and interactions. These effects can be probed by comparing the measured and predicted values of \(m_W\). In the context of global fits to the SM parameters, constraints on physics beyond the SM are currently limited by the W-boson mass measurement precision [16]. Improving the precision of the measurement of \(m_W\) is therefore of high importance for testing the overall consistency of the SM.

Previous measurements of the mass of the W boson were performed at the CERN SPS proton–antiproton (\(p\bar{p}\) ) collider with the UA1 and UA2 experiments [17, 18] at centre-of-mass energies of \(\sqrt{s}=546\,\text {GeV}\) and \(\sqrt{s}=630\,\text {GeV}\), at the Tevatron \(p\bar{p}\) collider with the CDF and D0 detectors at \(\sqrt{s}=1.8\,\text {TeV}\) [19, 20, 21] and \(\sqrt{s}=1.96\,\text {TeV}\) [22, 23, 24], and at the LEP electron–positron collider by the ALEPH, DELPHI, L3, and OPAL collaborations at \(\sqrt{s}=161\)\(209\,\text {GeV}\) [25, 26, 27, 28]. The current Particle Data Group world average value of \(m_W = 80385 \pm 15\) \(\,\text {MeV}\) [29] is dominated by the CDF and D0 measurements performed at \(\sqrt{s}=1.96\,\text {TeV}\). Given the precisely measured values of \(\alpha \), \(G_{\mu }\) and \(m_Z\), and taking recent top-quark and Higgs-boson mass measurements, the SM prediction of \(m_W\) is \(m_W=80358\pm 8\) MeV in Ref. [16] and \(m_W=80362\pm 8\) \(\,\text {MeV}\) in Ref. [30]. The SM prediction uncertainty of 8 \(\,\text {MeV}\) represents a target for the precision of future measurements of \(m_W\).

At hadron colliders, the W-boson mass can be determined in Drell–Yan production [31] from \(W\rightarrow \ell \nu \) decays, where \(\ell \) is an electron or muon. The mass of the W boson is extracted from the Jacobian edges of the final-state kinematic distributions, measured in the plane perpendicular to the beam direction. Sensitive observables include the transverse momenta of the charged lepton and neutrino and the W-boson transverse mass.

The ATLAS and CMS experiments benefit from large signal and calibration samples. The numbers of selected W- and Z-boson events, collected in a sample corresponding to approximately 4.6 fb\(^{-1}\) of integrated luminosity at a centre-of-mass energy of \(7\,\text {TeV}\), are of the order of \(10^7\) for the \(W\rightarrow \ell \nu \), and of the order of \(10^6\) for the \(Z\rightarrow \ell \ell \) processes. The available data sample is therefore larger by an order of magnitude compared to the corresponding samples used for the CDF and D0 measurements. Given the precisely measured value of the Z-boson mass [32] and the clean leptonic final state, the \(Z\rightarrow \ell \ell \) processes provide the primary constraints for detector calibration, physics modelling, and validation of the analysis strategy. The sizes of these samples correspond to a statistical uncertainty smaller than 10 \(\,\text {MeV}\) in the measurement of the W-boson mass.

Measurements of \(m_W\) at the LHC are affected by significant complications related to the strong interaction. In particular, in proton–proton (pp) collisions at \(\sqrt{s}=7\) \(\text {TeV}\), approximately 25% of the inclusive W-boson production rate is induced by at least one second-generation quark, s or c, in the initial state. The amount of heavy-quark-initiated production has implications for the W-boson rapidity and transverse-momentum distributions [33]. As a consequence, the measurement of the W-boson mass is sensitive to the strange-quark and charm-quark parton distribution functions (PDFs) of the proton. In contrast, second-generation quarks contribute only to approximately 5% of the overall W-boson production rate at the Tevatron. Other important aspects of the measurement of the W-boson mass are the theoretical description of electroweak corrections, in particular the modelling of photon radiation from the W- and Z-boson decay leptons, and the modelling of the relative fractions of helicity cross sections in the Drell–Yan processes [34].

This paper is structured as follows. Section 2 presents an overview of the measurement strategy. Section 3 describes the ATLAS detector. Section 4 describes the data and simulation samples used for the measurement. Section 5 describes the object reconstruction and the event selection. Section 6 summarises the modelling of vector-boson production and decay, with emphasis on the QCD effects outlined above. Sections 7 and 8 are dedicated to the electron, muon, and recoil calibration procedures. Section 9 presents a set of validation tests of the measurement procedure, performed using the Z-boson event sample. Section 10 describes the analysis of the W-boson sample. Section 11 presents the extraction of \(m_W\). The results are summarised in Sect. 12.

2 Measurement overview

This section provides the definition of the observables used in the analysis, an overview of the measurement strategy for the determination of the mass of the W boson, and a description of the methodology used to estimate the systematic uncertainties.

2.1 Observable definitions

ATLAS uses a right-handed coordinate system with its origin at the nominal interaction point (IP) in the centre of the detector and the z-axis along the beam pipe. The x-axis points from the IP to the centre of the LHC ring, and the y-axis points upward. Cylindrical coordinates \((r,\phi )\) are used in the transverse plane, \(\phi \) being the azimuth around the z-axis. The pseudorapidity is defined in terms of the polar angle \(\theta \) as \(\eta =-\ln \tan (\theta /2)\).

The kinematic properties of charged leptons from W- and Z-boson decays are characterised by the measured transverse momentum, \(p_{\text {T}} ^{\ell }\), pseudorapidity, \(\eta _{\ell }\), and azimuth, \(\phi _{\ell }\). The mass of the lepton, \(m_{\ell }\), completes the four-vector. For Z-boson events, the invariant mass, \(m_{\ell \ell }\), the rapidity, \(y_{\ell \ell }\), and the transverse momentum, \(p_{\text {T}} ^{\ell \ell }\), are obtained by combining the four-momenta of the decay-lepton pair.

The recoil in the transverse plane, \(\vec {u}_{\mathrm {T}}\), is reconstructed from the vector sum of the transverse energy of all clusters reconstructed in the calorimeters (Sect. 3), excluding energy deposits associated with the decay leptons. It is defined as:
$$\begin{aligned} \vec {u}_{\mathrm {T}}= \sum _i \vec {E}_{\mathrm {T},i}, \end{aligned}$$
where \(\vec {E}_{\mathrm {T},i}\) is the vector of the transverse energy of cluster i. The transverse-energy vector of a cluster has magnitude \(E_{\mathrm {T}} = E / \cosh \eta \), with the energy deposit of the cluster E and its pseudorapidity \(\eta \). The azimuth \(\phi \) of the transverse-energy vector is defined from the coordinates of the cluster in the transverse plane. In W- and Z-boson events, \(-\vec {u}_{\mathrm {T}}\) provides an estimate of the boson transverse momentum. The related quantities \(u_x\) and \(u_y\) are the projections of the recoil onto the axes of the transverse plane in the ATLAS coordinate system. In Z-boson events, \(u_{\parallel }^Z\) and \(u_{\perp }^Z\) represent the projections of the recoil onto the axes parallel and perpendicular to the Z-boson transverse momentum reconstructed from the decay-lepton pair. Whereas \(u_{\parallel }^Z\) can be compared to \(-p_{\mathrm {T}}^{\ell \ell }\) and probes the detector response to the recoil in terms of linearity and resolution, the \(u_{\perp }^Z\) distribution satisfies \(\left\langle u_{\perp }^Z \right\rangle =0\) and its width provides an estimate of the recoil resolution. In W-boson events, \(u_{\parallel }^\ell \) and \(u_{\perp }^\ell \) are the projections of the recoil onto the axes parallel and perpendicular to the reconstructed charged-lepton transverse momentum.

The resolution of the recoil is affected by additional event properties, namely the per-event number of pp interactions per bunch crossing (pile-up) \({\mu }\), the average number of pp interactions per bunch crossing \(\left\langle \mu \right\rangle \), the total reconstructed transverse energy, defined as the scalar sum of the transverse energy of all calorimeter clusters, \(\Sigma E_{\mathrm {T}} \equiv \sum _{i} E_{{\mathrm {T}},i}\), and the quantity \(\Sigma E^{*}_{\mathrm {T}} \equiv \Sigma E_{\mathrm {T}} - |\vec u_{\mathrm {T}}|\). The latter is less correlated with the recoil than \(\Sigma E_{\mathrm {T}}\), and better represents the event activity related to the pile-up and to the underlying event.

The magnitude and direction of the transverse-momentum vector of the decay neutrino, \(\vec {p}_\text {T}^{\,\nu }\), are inferred from the vector of the missing transverse momentum, \(\vec {p}_{\text {T}}^{\,\text {miss}} \), which corresponds to the momentum imbalance in the transverse plane and is defined as:
$$\begin{aligned} \vec {p}_{\text {T}}^{\,\text {miss}} = -\left( \vec {p}_{\mathrm {T}}^{\,\ell } + \vec {u}_{\mathrm {T}}\right) . \end{aligned}$$
The W-boson transverse mass, \(m_{\mathrm {T}}\), is derived from \(p_{\text {T}}^{\text {miss}} \) and from the transverse momentum of the charged lepton as follows:
$$\begin{aligned} m_{\mathrm {T}}= \sqrt{2 p_{\text {T}} ^\ell p_{\text {T}}^{\text {miss}} (1-\cos {\Delta \phi })}, \end{aligned}$$
where \(\Delta \phi \) is the azimuthal opening angle between the charged lepton and the missing transverse momentum.
All vector-boson masses and widths are defined in the running-width scheme. Resonances are expressed by the relativistic Breit–Wigner mass distribution:
$$\begin{aligned} \frac{\text {d}\sigma }{\text {d}m} \propto \frac{m^2}{(m^2-m_V^2)^2+m^4\Gamma _V^2/m_V^2}, \end{aligned}$$
(1)
where m is the invariant mass of the vector-boson decay products, and \(m_V\) and \(\Gamma _V\), with \(V = W,Z\), are the vector-boson masses and widths, respectively. This scheme was introduced in Ref. [35], and is consistent with earlier measurements of the W- and Z-boson resonance parameters [24, 32].

2.2 Analysis strategy

The mass of the W boson is determined from fits to the transverse momentum of the charged lepton, \(p_{\text {T}} ^\ell \), and to the transverse mass of the W boson, \(m_{\mathrm {T}}\). For W bosons at rest, the transverse-momentum distributions of the W decay leptons have a Jacobian edge at a value of m / 2, whereas the distribution of the transverse mass has an endpoint at the value of m [36], where m is the invariant mass of the charged-lepton and neutrino system, which is related to \(m_W\) through the Breit–Wigner distribution of Eq. (1).

The expected final-state distributions, referred to as templates, are simulated for several values of \(m_W\) and include signal and background contributions. The templates are compared to the observed distribution by means of a \(\chi ^2\) compatibility test. The \(\chi ^2\) as a function of \(m_W\) is interpolated, and the measured value is determined by analytical minimisation of the \(\chi ^2\) function. Predictions for different values of \(m_W\) are obtained from a single simulated reference sample, by reweighting the W-boson invariant mass distribution according to the Breit–Wigner parameterisation of Eq. (1). The W-boson width is scaled accordingly, following the SM relation \(\Gamma _W \propto m_W^3\).

Experimentally, the \(p_{\text {T}} ^\ell \) and \(p_{\text {T}}^{\text {miss}} \) distributions are affected by the lepton energy calibration. The latter is also affected by the calibration of the recoil. The \(p_{\text {T}} ^\ell \) and \(p_{\text {T}}^{\text {miss}}\) distributions are broadened by the W-boson transverse-momentum distribution, and are sensitive to the W-boson helicity states, which are influenced by the proton PDFs [37]. Compared to \(p_{\text {T}} ^\ell \), the \(m_{\mathrm {T}}\) distribution has larger uncertainties due to the recoil, but smaller sensitivity to such physics-modelling effects. Imperfect modelling of these effects can distort the template distributions, and constitutes a significant source of uncertainties for the determination of \(m_W\).

The calibration procedures described in this paper rely mainly on methods and results published earlier by ATLAS [38, 39, 40], and based on W and Z samples at \(\sqrt{s}=7\) \(\text {TeV}\) and \(\sqrt{s}=8\,\text {TeV}\). The \(Z\rightarrow \ell \ell \) event samples are used to calibrate the detector response. Lepton momentum corrections are derived exploiting the precisely measured value of the Z-boson mass, \(m_Z\) [32], and the recoil response is calibrated using the expected momentum balance with \(p_{\mathrm {T}}^{\ell \ell }\). Identification and reconstruction efficiency corrections are determined from W- and Z-boson events using the tag-and-probe method [38, 40]. The dependence of these corrections on \(p_{\text {T}} ^\ell \) is important for the measurement of \(m_W\), as it affects the shape of the template distributions.

The detector response corrections and the physics modelling are verified in Z-boson events by performing measurements of the Z-boson mass with the same method used to determine the W-boson mass, and comparing the results to the LEP combined value of \(m_Z\), which is used as input for the lepton calibration. The determination of \(m_Z\) from the lepton-pair invariant mass provides a first closure test of the lepton energy calibration. In addition, the extraction of \(m_Z\) from the \(p_{\text {T}} ^\ell \) distribution tests the \(p_{\text {T}} ^\ell \)-dependence of the efficiency corrections, and the modelling of the Z-boson transverse-momentum distribution and of the relative fractions of Z-boson helicity states. The \(p_{\text {T}}^{\text {miss}}\) and \(m_{\mathrm {T}}\) variables are defined in Z-boson events by treating one of the reconstructed decay leptons as a neutrino. The extraction of \(m_Z\) from the \(m_{\mathrm {T}}\) distribution provides a test of the recoil calibration. The combination of the extraction of \(m_Z\) from the \(m_{\ell \ell }\), \(p_{\text {T}} ^\ell \) and \(m_{\mathrm {T}}\) distributions provides a closure test of the measurement procedure. The precision of this validation procedure is limited by the finite size of the Z-boson sample, which is approximately ten times smaller than the W-boson sample.
Table 1

Summary of categories and kinematic distributions used in the \(m_W\) measurement analysis for the electron and muon decay channels

Decay channel

\(W\rightarrow e\nu \)

\(W\rightarrow \mu \nu \)

Kinematic distributions

\(p_{\text {T}} ^\ell \), \(m_{\mathrm {T}}\)

\(p_{\text {T}} ^\ell \), \(m_{\mathrm {T}}\)

Charge categories

\(W^+\), \(W^-\)

\(W^+\), \(W^-\)

\(|\eta _\ell |\) categories

[0, 0.6], [0.6, 1.2], [1.8, 2.4]

[0, 0.8], [0.8, 1.4], [1.4, 2.0], [2.0, 2.4]

The analysis of the Z-boson sample does not probe differences in the modelling of W- and Z-boson production processes. Whereas W-boson production at the Tevatron is charge symmetric and dominated by interactions with at least one valence quark, the sea-quark PDFs play a larger role at the LHC, and contributions from processes with heavy quarks in the initial state have to be modelled properly. The \(W^+\)-boson production rate exceeds that of \(W^-\) bosons by about 40%, with a broader rapidity distribution and a softer transverse-momentum distribution. Uncertainties in the modelling of these distributions and in the relative fractions of the W-boson helicity states are constrained using measurements of W- and Z-boson production performed with the ATLAS experiment at \(\sqrt{s}=7\) \(\text {TeV}\) and \(\sqrt{s}=8\) \(\text {TeV}\) [41, 42, 43, 44, 45].

The final measured value of the W-boson mass is obtained from the combination of various measurements performed in the electron and muon decay channels, and in charge- and \(|\eta _\ell |\)-dependent categories, as defined in Table 1. The boundaries of the \(|\eta _\ell |\) categories are driven mainly by experimental and statistical constraints. The measurements of \(m_W\) used in the combination are based on the observed distributions of \(p_{\text {T}} ^\ell \) and \(m_{\mathrm {T}}\), which are only partially correlated. Measurements of \(m_W\) based on the \(p_{\text {T}}^{\text {miss}}\) distributions are performed as consistency tests, but they are not used in the combination due to their significantly lower precision. The consistency of the results in the electron and muon channels provide a further test of the experimental calibrations, whereas the consistency of the results for the different charge and \(|\eta _\ell |\) categories tests the W-boson production model.

Further consistency tests are performed by repeating the measurement in three intervals of \(\left\langle \mu \right\rangle \), in two intervals of \(u_{\mathrm {T}}\) and \(u_{\parallel }^\ell \), and by removing the \(p_{\text {T}}^{\text {miss}}\) selection requirement, which is applied in the nominal signal selection. The consistency of the values of \(m_W\) in these additional categories probes the modelling of the recoil response, and the modelling of the transverse-momentum spectrum of the W boson. Finally, the stability of the result with respect to the charged-lepton azimuth, and upon variations of the fitting ranges is verified.

Systematic uncertainties in the determination of \(m_W\) are evaluated using pseudodata samples produced from the nominal simulated event samples by varying the parameters corresponding to each source of uncertainty in turn. The differences between the values of \(m_W\) extracted from the pseudodata and nominal samples are used to estimate the uncertainty. When relevant, these variations are applied simultaneously in the W-boson signal samples and in the background contributions. The systematic uncertainties are estimated separately for each source and for fit ranges of \(32<p_{\text {T}} ^\ell <45\,\text {GeV}\) and \(66<m_{\mathrm {T}}<99\,\text {GeV}\). These fit ranges minimise the total expected measurement uncertainty, and are used for the final result as discussed in Sect. 11.

In Sects. 6, 7, 8, and 10, which discuss the systematic uncertainties of the \(m_W\) measurement, the uncertainties are also given for combinations of measurement categories. This provides information showing the reduction of the systematic uncertainty obtained from the measurement categorisation. For these cases, the combined uncertainties are evaluated including only the expected statistical uncertainty in addition to the systematic uncertainty being considered. However, the total measurement uncertainty is estimated by adding all uncertainty contributions in quadrature for each measurement category, and combining the results accounting for correlations across categories.

During the analysis, an unknown offset was added to the value of \(m_W\) used to produce the templates. The offset was randomly selected from a uniform distribution in the range \([-100,100]\) \(\,\text {MeV}\), and the same value was used for the \(W^{+}\) and \(W^{-}\) templates. The offset was removed after the \(m_W\) measurements performed in all categories were found to be compatible and the analysis procedure was finalised.

3 The ATLAS detector

The ATLAS experiment [46] is a multipurpose particle detector with a forward-backward symmetric cylindrical geometry. It consists of an inner tracking detector surrounded by a thin superconducting solenoid, electromagnetic and hadronic calorimeters, and a muon spectrometer incorporating three large superconducting toroid magnets.

The inner-detector system (ID) is immersed in a 2 T axial magnetic field and provides charged-particle tracking in the range \(|\eta | < 2.5\). At small radii, a high-granularity silicon pixel detector covers the vertex region and typically provides three measurements per track. It is followed by the silicon microstrip tracker, which usually provides eight measurement points per track. These silicon detectors are complemented by a gas-filled straw-tube transition radiation tracker, which enables radially extended track reconstruction up to \(|\eta | = 2.0\). The transition radiation tracker also provides electron identification information based on the fraction of hits (typically 35 in total) above a higher energy-deposit threshold corresponding to transition radiation.

The calorimeter system covers the pseudorapidity range \(|\eta | < 4.9\). Within the region \(|\eta |< 3.2\), electromagnetic (EM) calorimetry is provided by high-granularity lead/liquid-argon (LAr) calorimeters, with an additional thin LAr presampler covering \(|\eta |<1.8\) to correct for upstream energy-loss fluctuations. The EM calorimeter is divided into a barrel section covering \(|\eta |<1.475\) and two endcap sections covering \(1.375<|\eta |<3.2\). For \(|\eta |<2.5\) it is divided into three layers in depth, which are finely segmented in \(\eta \) and \(\phi \). Hadronic calorimetry is provided by a steel/scintillator-tile calorimeter, segmented into three barrel structures within \(|\eta | < 1.7\) and two copper/LAr hadronic endcap calorimeters covering \(1.5<|\eta |<3.2\). The solid-angle coverage is completed with forward copper/LAr and tungsten/LAr calorimeter modules in \(3.1<|\eta |<4.9\), optimised for electromagnetic and hadronic measurements, respectively.

The muon spectrometer (MS) comprises separate trigger and high-precision tracking chambers measuring the deflection of muons in a magnetic field generated by superconducting air-core toroids. The precision chamber system covers the region \(|\eta | < 2.7\) with three layers of monitored drift tubes, complemented by cathode strip chambers in the forward region. The muon trigger system covers the range \(|\eta | < 2.4\) with resistive plate chambers in the barrel, and thin gap chambers in the endcap regions.

A three-level trigger system is used to select events for offline analysis [47]. The level-1 trigger is implemented in hardware and uses a subset of detector information to reduce the event rate to a design value of at most 75 kHz. This is followed by two software-based trigger levels which together reduce the event rate to about 300 Hz.

4 Data samples and event simulation

The data sample used in this analysis consists of W- and Z-boson candidate events, collected in 2011 with the ATLAS detector in proton–proton collisions at the LHC, at a centre-of-mass energy of \(\sqrt{s}=7\) \(\text {TeV}\). The sample for the electron channel, with all relevant detector systems operational, corresponds to approximately 4.6 fb\(^{-1}\) of integrated luminosity. A smaller integrated luminosity of approximately 4.1 fb\(^{-1}\) is used in the muon channel, as part of the data was discarded due to a timing problem in the resistive plate chambers, which affected the muon trigger efficiency. The relative uncertainty of the integrated luminosity is 1.8% [48]. This data set provides approximately 1.4 \(\times 10^7\) reconstructed W-boson events and 1.8 \(\times 10^6\) Z-boson events, after all selection criteria have been applied.

The Powheg MC generator [49, 50, 51] (v1/r1556) is used for the simulation of the hard-scattering processes of W- and Z-boson production and decay in the electron, muon, and tau channels, and is interfaced to Pythia 8 (v8.170) for the modelling of the parton shower, hadronisation, and underlying event [52, 53], with parameters set according to the AZNLO tune [44]. The CT10 PDF set [54] is used for the hard-scattering processes, whereas the CTEQ6L1 PDF set [55] is used for the parton shower. In the Z-boson samples, the effect of virtual photon production (\(\gamma ^*\)) and \(Z/\gamma ^*\) interference is included. The effect of QED final-state radiation (FSR) is simulated with Photos (v2.154) [56]. Tau lepton decays are handled by Pythia 8, taking into account polarisation effects. An alternative set of samples for W- and Z-boson production is generated with Powheg interfaced to Herwig (v6.520) for the modelling of the parton shower [57], and to Jimmy (v4.31) for the underlying event [58]. The W- and Z-boson masses are set to \(m_W=80.399\,\text {GeV}\) and \(m_Z=91.1875\,\text {GeV}\), respectively. During the analysis, the value of the W-boson mass in the \(W\rightarrow \ell \nu \) and \(W\rightarrow \tau \nu \) samples was blinded using the reweighting procedure described in Sect. 2.

Top-quark pair production and the single-top-quark processes are modelled using the MC@NLO MC generator (v4.01) [59, 60, 61], interfaced to Herwig and Jimmy. Gauge-boson pair production (WW, WZ, ZZ) is simulated with Herwig (v6.520). In all the samples, the CT10 PDF set is used. Samples of heavy-flavour multijet events (\(pp\rightarrow b\bar{b} +X\) and \(pp\rightarrow c \bar{c} +X\)) are simulated with Pythia 8 to validate the data-driven methods used to estimate backgrounds with non-prompt leptons in the final state.

Whereas the extraction of \(m_W\) is based on the shape of distributions, and is not sensitive to the overall normalisation of the predicted distributions, it is affected by theoretical uncertainties in the relative fractions of background and signal. The W- and Z-boson event yields are normalised according to their measured cross sections, and uncertainties of 1.8% and 2.3% are assigned to the \(W^{+}/Z\) and \(W^{-}/Z\) production cross-section ratios, respectively [41]. The \(t\bar{t} \) sample is normalised according to its measured cross section [62] with an uncertainty of 3.9%, whereas the cross-section predictions for the single-top production processes of Refs. [63, 64, 65] are used for the normalisation of the corresponding sample, with an uncertainty of 7%. The samples of events with massive gauge-boson pair production are normalised to the NLO predictions calculated with MCFM [66], with an uncertainty of 10% to cover the differences to the NNLO predictions [67].

The response of the ATLAS detector is simulated using a program [68] based on Geant 4 [69]. The ID and the MS were simulated assuming an ideal detector geometry; alignment corrections are applied to the data during event reconstruction. The description of the detector material incorporates the results of extensive studies of the electron and photon calibration [39]. The simulated hard-scattering process is overlaid with additional proton–proton interactions, simulated with Pythia 8 (v8.165) using the A2 tune [70]. The distribution of the average number of interactions per bunch crossing \(\left\langle \mu \right\rangle \) spans the range 2.5–16.0, with a mean value of approximately 9.0.

Simulation inaccuracies affecting the distributions of the signal, the response of the detector, and the underlying-event modelling, are corrected as described in the following sections. Physics-modelling corrections, such as those affecting the W-boson transverse-momentum distribution and the angular decay coefficients, are discussed in Sect. 6. Calibration and detector response corrections are presented in Sects. 7 and 8.

5 Particle reconstruction and event selection

This section describes the reconstruction and identification of electrons and muons, the reconstruction of the recoil, and the requirements used to select W- and Z-boson candidate events. The recoil provides an event-by-event estimate of the W-boson transverse momentum. The reconstructed kinematic properties of the leptons and of the recoil are used to infer the transverse momentum of the neutrino and the transverse-mass kinematic variables.

5.1 Reconstruction of electrons, muons and the recoil

Electron candidates are reconstructed from clusters of energy deposited in the electromagnetic calorimeter and associated with at least one track in the ID [38, 39]. Quality requirements are applied to the associated tracks in order to reject poorly reconstructed charged-particle trajectories. The energy of the electron is reconstructed from the energy collected in calorimeter cells within an area of size \(\Delta \eta \times \Delta \phi = 0.075\times 0.175\) in the barrel, and \(0.125\times 0.125\) in the endcaps. A multivariate regression algorithm, developed and optimised on simulated events, is used to calibrate the energy reconstruction. The reconstructed electron energy is corrected to account for the energy deposited in front of the calorimeter and outside the cluster, as well as for variations of the energy response as a function of the impact point of the electron in the calorimeter. The energy calibration algorithm takes as inputs the energy collected by each calorimeter layer, including the presampler, the pseudorapidity of the cluster, and the local position of the shower within the cell of the second layer, which corresponds to the cluster centroid. The kinematic properties of the reconstructed electron are inferred from the energy measured in the EM calorimeter, and from the pseudorapidity and azimuth of the associated track. Electron candidates are required to have \(p_{\text {T}} > 15\,\text {GeV}\) and \(|\eta |<2.4\) and to fulfil a set of tight identification requirements [38]. The pseudorapidity range \(1.2<|\eta |<1.82\) is excluded from the measurement, as the amount of passive material in front of the calorimeter and its uncertainty are largest in this region [39], preventing a sufficiently accurate description of non-Gaussian tails in the electron energy response. Additional isolation requirements on the nearby activity in the ID and calorimeter are applied to improve the background rejection. These isolation requirements are implemented by requiring the scalar sum of the \(p_{\text {T}}\) of tracks in a cone of size \(\Delta R \equiv \sqrt{(\Delta \eta )^2+(\Delta \phi )^2} < 0.4\) around the electron, \(p_{\text {T}} ^{e,\text {cone}}\), and the transverse energy deposited in the calorimeter within a cone of size \(\Delta R <0.2\) around the electron, \(E_\text {T}^\text {cone}\), to be small. The contribution from the electron candidate itself is excluded. The specific criteria are optimised as a function of electron \(\eta \) and \(p_{\text {T}}\) to have a combined efficiency of about 95% in the simulation for isolated electrons from the decay of a W or Z boson.

The muon reconstruction is performed independently in the ID and in the MS, and a combined muon candidate is formed from the combination of a MS track with an ID track, based on the statistical combination of the track parameters [40]. The kinematic properties of the reconstructed muon are defined using the ID track parameters alone, which allows a simpler calibration procedure. The loss of resolution is small (10–15%) in the transverse-momentum range relevant for the measurement of the W-boson mass. The ID tracks associated with the muons must satisfy quality requirements on the number of hits recorded by each subdetector [40]. In order to reject muons from cosmic rays, the longitudinal coordinate of the point of closest approach of the track to the beamline is required to be within 10 mm of the collision vertex. Muon candidates are required to have \(p_{\text {T}} >20\,\text {GeV}\) and \(|\eta |<2.4\). Similarly to the electrons, the rejection of multijet background is increased by applying an isolation requirement : the scalar sum of the \(p_{\text {T}}\) of tracks in a cone of size \(\Delta R < 0.2\) around the muon candidate, \(p_{\text {T}} ^{\mu ,\text {cone}}\), is required to be less than 10% of the muon \(p_{\text {T}}\).

The recoil, \(\vec {u}_{\mathrm {T}}\), is reconstructed from the vector sum of the transverse energy of all clusters measured in the calorimeters, as defined in Sect. 2.1. The ATLAS calorimeters measure energy depositions in the range \(|\eta |<4.9\) with a topological clustering algorithm [71], which starts from cells with an energy of at least four times the expected noise from electronics and pile-up. The momentum vector of each cluster is determined by the magnitude and coordinates of the energy deposition. Cluster energies are initially measured assuming that the energy deposition occurs only through electromagnetic interactions, and are then corrected for the different calorimeter responses to hadrons and electromagnetic particles, for losses due to dead material, and for energy which is not captured by the clustering process. The definition of \(\vec {u}_{\mathrm {T}}\) and the inferred quantities \(p_{\text {T}}^{\text {miss}} \) and \(m_{\mathrm {T}}\) do not involve the explicit reconstruction of particle jets, to avoid possible threshold effects.

Clusters located a distance \(\Delta R < 0.2\) from the reconstructed electron or muon candidates are not used for the reconstruction of \(\vec {u}_{\mathrm {T}}\). This ensures that energy deposits originating from the lepton itself or from accompanying photons (from FSR or Bremsstrahlung) do not contribute to the recoil measurement. The energy of any soft particles removed along with the lepton is compensated for using the total transverse energy measured in a cone of the same size \(\Delta R =0.2\), placed at the same absolute pseudorapidity as the lepton with randomly chosen sign, and at different \(\phi \). The total transverse momentum measured in this cone is rotated to the position of the lepton and added to \(\vec {u}_{\mathrm {T}}\).

5.2 Event selection

The W-boson sample is collected during data-taking with triggers requiring at least one muon candidate with transverse momentum larger than \(18\,\text {GeV}\) or at least one electron candidate with transverse momentum larger than \(20\,\text {GeV}\). The transverse-momentum requirement for the electron candidate was raised to \(22\,\text {GeV}\) in later data-taking periods to cope with the increased instantaneous luminosity delivered by the LHC. Selected events are required to have a reconstructed primary vertex with at least three associated tracks.

W-boson candidate events are selected by requiring exactly one reconstructed electron or muon with \(p_{\text {T}} ^\ell > 30\,\text {GeV}\). The leptons are required to match the corresponding trigger object. In addition, the reconstructed recoil is required to be \(u_{\mathrm {T}}< 30\,\text {GeV}\), the missing transverse momentum \(p_{\text {T}}^{\text {miss}} > 30\,\text {GeV}\) and the transverse mass \(m_{\mathrm {T}}> 60\,\text {GeV}\). These selection requirements are optimised to reduce the multijet background contribution, and to minimise model uncertainties from W bosons produced at high transverse momentum. A total of 5.89 \(\times 10^6\) W-boson candidate events are selected in the \(W\rightarrow e\nu \) channel, and 7.84 \(\times 10^6\) events in the \(W\rightarrow \mu \nu \) channel.

As mentioned in Sect. 2, Z-boson events are extensively used to calibrate the response of the detector to electrons and muons, and to derive recoil corrections. In addition, Z-boson events are used to test several aspects of the modelling of vector-boson production. Z-boson candidate events are collected with the same trigger selection used for the W-boson sample. The analysis selection requires exactly two reconstructed leptons with \(p_{\text {T}} ^\ell > 25\,\text {GeV}\), having the same flavour and opposite charges. The events are required to have an invariant mass of the dilepton system in the range \(80<m_{\ell \ell }<100\,\text {GeV}\). In both channels, selected leptons are required to be isolated in the same way as in the W-boson event selection. In total, 0.58 \(\times 10^6\) and 1.23 \(\times 10^6\) Z-boson candidate events are selected in the electron and muon decay channels, respectively.

6 Vector-boson production and decay

Samples of inclusive vector-boson production are produced using the Powheg MC generator interfaced to Pythia 8, henceforth referred to as Powheg+Pythia 8. The W- and Z-boson samples are reweighted to include the effects of higher-order QCD and electroweak (EW) corrections, as well as the results of fits to measured distributions which improve the agreement of the simulated lepton kinematic distributions with the data. The effect of virtual photon production and \(Z/\gamma ^*\) interference is included in both the predictions and the Powheg+Pythia 8 simulated Z-boson samples. The reweighting procedure used to include the corrections in the simulated event samples is detailed in Sect. 6.4.

The correction procedure is based on the factorisation of the fully differential leptonic Drell–Yan cross section [31] into four terms:
$$\begin{aligned} \frac{\text {d}\sigma }{\text {d}p_1 \, \text {d}p_2}= & {} \left[ \frac{\text {d}\sigma (m)}{\text {d}m}\right] \left[ \frac{\text {d}\sigma (y)}{\text {d}y}\right] \left[ \frac{\text {d}\sigma (p_{\text {T}}, y)}{\text {d}p_{\text {T}} \,\text {d}y} \left( \frac{\text {d}\sigma (y)}{\text {d}y}\right) ^{-1} \right] \nonumber \\&\times \left[ (1{+}\cos ^2\theta ){+}\sum _{i=0}^{7} A_i(p_{\text {T}},y) P_i(\cos \theta , \phi ) \right] ,\nonumber \\ \end{aligned}$$
(2)
where \(p_1\) and \(p_2\) are the lepton and anti-lepton four-momenta; m, \(p_{\text {T}}\), and y are the invariant mass, transverse momentum, and rapidity of the dilepton system; \(\theta \) and \(\phi \) are the polar angle and azimuth of the lepton1 in any given rest frame of the dilepton system; \(A_i\) are numerical coefficients, and \(P_i\) are spherical harmonics of order zero, one and two.
The differential cross section as a function of the invariant mass, \(\text {d}\sigma (m)/\text {d}m\), is modelled with a Breit–Wigner parameterisation according to Eq. (1). In the case of the Z-boson samples, the photon propagator is included using the running electromagnetic coupling constant; further electroweak corrections are discussed in Sect. 6.1. The differential cross section as a function of boson rapidity, \(\text {d}\sigma (y)/\text {d}y\), and the coefficients \(A_i\) are modelled with perturbative QCD fixed-order predictions, as described in Sect. 6.2. The transverse-momentum spectrum at a given rapidity, \(\text {d}\sigma (p_{\text {T}},y)/(\text {d}p_{\text {T}} \,\text {d}y) \cdot (\text {d}\sigma (y)/\text {d}y)^{-1}\), is modelled with predictions based on the Pythia  8 MC generator, as discussed in Sect. 6.3. An exhaustive review of available predictions for W- and Z-boson production at the LHC is given in Ref. [72].
Table 2

Impact on the \(m_W\) measurement of systematic uncertainties from higher-order electroweak corrections, for the \(p_{\text {T}} ^\ell \) and \(m_{\mathrm {T}}\) distributions in the electron and muon decay channels

Decay channel

\(W \rightarrow e \nu \)

\(W \rightarrow \mu \nu \)

Kinematic distribution

\(p_{\text {T}} ^\ell \)

\(m_{\mathrm {T}}\)

\(p_{\text {T}} ^\ell \)

\(m_{\mathrm {T}}\)

\(\delta m_W\) [MeV]

     FSR (real)

\(<0.1\)

\(<0.1\)

\(<0.1\)

\(<0.1\)

     Pure weak and IFI corrections

3.3

2.5

3.5

2.5

     FSR (pair production)

3.6

0.8

4.4

0.8

     Total

4.9

2.6

5.6

2.6

Measurements of \(W\)- and \(Z\)-boson production are used to validate and constrain the modelling of the fully differential leptonic Drell–Yan cross section. The PDF central values and uncertainties, as well as the modelling of the differential cross section as a function of boson rapidity, are validated by comparing to the 7 \(\text {TeV}\) \(W\)- and \(Z\)-boson rapidity measurements [41], based on the same data sample. The QCD parameters of the parton shower model were determined by fits to the transverse-momentum distribution of the Z boson measured at 7 \(\text {TeV}\) [44]. The modelling of the \(A_i\) coefficients is validated by comparing the theoretical predictions to the 8 \(\text {TeV}\) measurement of the angular coefficients in Z-boson decays [42].

6.1 Electroweak corrections and uncertainties

The dominant source of electroweak corrections to \(W\)- and \(Z\)-boson production originates from QED final-state radiation, and is simulated with Photos. The effect of QED initial-state radiation (ISR) is also included through the Pythia 8 parton shower. The uncertainty in the modelling of QED FSR is evaluated by comparing distributions obtained using the default leading-order photon emission matrix elements with predictions obtained using NLO matrix elements, as well as by comparing Photos with an alternative implementation based on the Yennie–Frautschi–Suura formalism [73], which is available in Winhac [74]. The differences are small in both cases, and the associated uncertainty is considered negligible.

Other sources of electroweak corrections are not included in the simulated event samples, and their full effects are considered as systematic uncertainties. They include the interference between ISR and FSR QED corrections (IFI), pure weak corrections due to virtual-loop and box diagrams, and final-state emission of lepton pairs. Complete \(O(\alpha )\) electroweak corrections to the \(pp\rightarrow W+X\), \(W\rightarrow \ell \nu \) process were initially calculated in Refs. [75, 76]. Combined QCD and EW corrections are however necessary to evaluate the effect of the latter in presence of a realistic \(p_{\text {T}} ^W\) distribution. Approximate \(O(\alpha _{\mathrm s}\alpha )\) corrections including parton shower effects are available from Winhac, Sanc [77] and in the Powheg framework [78, 79, 80]. A complete, fixed-order calculation of \(O(\alpha _{\mathrm s}\alpha )\) corrections in the resonance region appeared in Ref. [81].

In the present work the effect of the NLO EW corrections are estimated using Winhac, which employs the Pythia 6 MC generator for the simulation of QCD and QED ISR. The corresponding uncertainties are evaluated comparing the final state distributions obtained including QED FSR only with predictions using the complete NLO EW corrections in the \(\alpha (0)\) and \(G_\mu \) renormalisation schemes [82]. The latter predicts the larger correction and is used to assign the systematic uncertainty.

Final-state lepton pair production, through \(\gamma ^*\rightarrow \ell \ell \) radiation, is formally a higher-order correction but constitutes an significant additional source of energy loss for the W-boson decay products. This process is not included in the event simulation, and the impact on the determination of \(m_W\) is evaluated using Photos and Sanc.

Table 2 summarises the effect of the uncertainties associated with the electroweak corrections on the \(m_W\) measurements. All comparisons described above were performed at particle level. The impact is larger for the \(p_{\text {T}} ^\ell \) distribution than for the \(m_{\mathrm {T}}\) distribution, and similar between the electron and muon decay channels. A detailed evaluation of these uncertainties was performed in Ref. [83] using Powheg [78], and the results are in fair agreement with Table 2. The study of Ref. [83] also compares, at fixed order, the effect of the approximate \(O(\alpha _{\mathrm s}\alpha )\) corrections with the full calculation of Ref. [81], and good agreement is found. The same sources of uncertainty affect the lepton momentum calibration through their impact on the \(m_{\ell \ell }\) distribution in Z-boson events, as discussed in Sect. 7.

6.2 Rapidity distribution and angular coefficients

At leading order, W and Z bosons are produced with zero transverse momentum, and the angular distribution of the decay leptons depends solely on the polar angle of the lepton in the boson rest frame. Higher-order corrections give rise to sizeable boson transverse momentum, and to azimuthal asymmetries in the angular distribution of the decay leptons. The angular distribution of the W- and Z-boson decay leptons is determined by the relative fractions of helicity cross sections for the vector-boson production. The fully differential leptonic Drell–Yan cross section can be decomposed as a weighted sum of nine harmonic polynomials, with weights given by the helicity cross sections. The harmonic polynomials depend on the polar angle, \(\theta \), and the azimuth, \(\phi \), of the lepton in a given rest frame of the boson. The helicity cross sections depend, in their most general expression, on the transverse momentum, \(p_{\text {T}} \), rapidity, y, and invariant mass, m, of the boson. It is customary to factorise the unpolarised, or angular-integrated, cross section, \(\text {d}\sigma /(\text {d}p_{\text {T}}^{2} \, \text {d}y \, \text {d}m)\), and express the decomposition in terms of dimensionless angular coefficients, \(A_{i}\), which represent the ratios of the helicity cross sections with respect to the unpolarised cross section [34], leading to the following expression for the fully differential Drell–Yan cross section:
$$\begin{aligned} \frac{\text {d}\sigma }{\text {d}p_{\text {T}} ^{2}\, \text {d}y\, \text {d}m\, \text {d}\cos \theta \, \text {d}\phi }= & {} \frac{3}{16\pi }\frac{\text {d}\sigma }{\text {d}p_{\text {T}} ^{2}\, \text {d}y\, \text {d}m} \nonumber \\&\times \left[ (1+\cos ^{2} \theta ) + A_{0} \, \frac{1}{2}(1-3\cos ^{2}\theta ) \right. \nonumber \\&+ A_{1} \, \sin 2\theta \cos \phi + A_{2} \, \frac{1}{2}\sin ^{2}\theta \cos 2\phi \nonumber \\&+ A_{3}\, \sin \theta \cos \phi + A_{4}\, \cos \theta \nonumber \\&+ A_{5}\, \sin ^{2}\theta \sin 2\phi + A_{6}\, \sin 2\theta \sin \phi \nonumber \\&\left. + A_{7}\, \sin \theta \sin \phi \right] . \end{aligned}$$
(3)
The angular coefficients depend in general on \(p_{\text {T}}\), y and m. The \(A_{5}\)\(A_{7}\) coefficients are non-zero only at order \(O(\alpha _{\mathrm s}^2)\) and above. They are small in the \(p_{\text {T}}\) region relevant for the present analysis, and are not considered further. The angles \(\theta \) and \(\phi \) are defined in the Collins–Soper (CS) frame [84].
The differential cross section as a function of boson rapidity, \(\text {d}\sigma (y)/\text {d}y\), and the angular coefficients, \(A_i\), are modelled with fixed-order perturbative QCD predictions, at \(O(\alpha _{\mathrm s}^2)\) in the perturbative expansion of the strong coupling constant and using the CT10nnlo PDF set [85]. The dependence of the angular coefficients on m is neglected; the effect of this approximation on the measurement of \(m_W\) is discussed in Sect. 6.4. For the calculation of the predictions, an optimised version of DYNNLO [86] is used, which explicitly decomposes the calculation of the cross section into the different pieces of the \(q_{\mathrm T}\)-subtraction formalism, and allows the computation of statistically correlated PDF variations. In this optimised version of DYNNLO, the Cuba library [87] is used for the numerical integration.
Fig. 1

a Normalised differential cross section as a function of \(p_{\text {T}} ^{\ell \ell }\) in Z-boson events [44] and b differential cross-section ratio \(R_{W/Z}(p_{\text {T}})\) as a function of the boson \(p_{\text {T}} \) [44, 45]. The measured cross sections are compared to the predictions of the Pythia 8 AZ tune and, in a, of the Pythia 8 4C tune. The shaded bands show the total experimental uncertainties

Fig. 2

Ratios of the reconstruction-level a \(p_{\text {T}} ^\ell \) and b \(m_{\mathrm {T}}\) normalised distributions obtained using Powheg+Pythia 8 AZNLO, DYRes and Powheg MiNLO+Pythia 8 to the baseline normalised distributions obtained using Pythia 8 AZ

The values of the angular coefficients predicted by the Powheg+Pythia 8 samples differ significantly from the corresponding NNLO predictions. In particular, large differences are observed in the predictions of \(A_0\) at low values of \(p_{\text {T}} ^{W,Z}\). Other coefficients, such as \(A_1\) and \(A_2\), are affected by significant NNLO corrections at high \(p_{\text {T}} ^{W,Z}\). In Z-boson production, \(A_3\) and \(A_4\) are sensitive to the vector couplings between the Z boson and the fermions, and are predicted assuming the measured value of the effective weak mixing angle \(\sin ^2\theta ^\ell _{\text {eff}}\) [32].

6.3 Transverse-momentum distribution

Predictions of the vector-boson transverse-momentum spectrum cannot rely solely on fixed-order perturbative QCD. Most \(W\)-boson events used for the analysis have a low transverse-momentum value, in the kinematic region \(p_{\text {T}} ^W < 30\,\text {GeV}\), where large logarithmic terms of the type \(\log (m_W/p_{\text {T}} ^W)\) need to be resummed, and non-perturbative effects must be included, either with parton showers or with predictions based on analytic resummation [88, 89, 90, 91, 92]. The modelling of the transverse-momentum spectrum of vector bosons at a given rapidity, expressed by the term \(\text {d}\sigma (p_{\text {T}},y)/(\text {d}p_{\text {T}} \,\text {d}y) \cdot (\text {d}\sigma (y)/\text {d}y)^{-1}\) in Eq. (2), is based on the Pythia 8 parton shower MC generator. The predictions of vector-boson production in the Pythia 8 MC generator employ leading-order matrix elements for the \(q\bar{q}'\rightarrow W, Z\) processes and include a reweighting of the first parton shower emission to the leading-order V+jet cross section [93]. The resulting prediction of the boson \(p_{\text {T}}\) spectrum is comparable in accuracy to those of an NLO plus parton shower generator setup such as Powheg+Pythia 8, and of resummed predictions at next-to-leading logarithmic order [94].

The values of the QCD parameters used in Pythia 8 were determined from fits to the Z-boson transverse momentum distribution measured with the ATLAS detector at a centre-of-mass energy of \(\sqrt{s} = 7\,\,\text {TeV}\) [44]. Three QCD parameters were considered in the fit: the intrinsic transverse momentum of the incoming partons, the value of \(\alpha _{\mathrm s}(m_Z)\) used for the QCD ISR, and the value of the ISR infrared cut-off. The resulting values of the Pythia  8 parameters constitute the AZ tune. The Pythia 8 AZ prediction was found to provide a satisfactory description of the \(p_{\text {T}} ^Z\) distribution as a function of rapidity, contrarily to Powheg+Pythia 8  AZNLO; hence the former is chosen to predict the \(p_{\text {T}} ^W\) distribution. The good consistency of the \(m_W\) measurement results in \(|\eta _\ell |\) categories, presented in Sect. 11, is also a consequence of this choice.

To illustrate the results of the parameters optimisation, the Pythia 8 AZ and 4C [95] predictions of the \(p_{\text {T}} ^Z\) distribution are compared in Fig. 1a to the measurement used to determine the AZ tune. Kinematic requirements on the decay leptons are applied according to the experimental acceptance. For further validation, the predicted differential cross-section ratio,
$$\begin{aligned} R_{W/Z}(p_{\text {T}}) = \left( \frac{1}{\sigma _W} \cdot \frac{\text {d}\sigma _W(p_{\text {T}})}{\text {d}p_{\text {T}}}\right) \left( \frac{1}{\sigma _Z} \cdot \frac{\text {d}\sigma _Z(p_{\text {T}})}{\text {d}p_{\text {T}}}\right) ^{-1}, \end{aligned}$$
is compared to the corresponding ratio of ATLAS measurements of vector-boson transverse momentum [44, 45]. The comparison is shown in Fig. 1b, where kinematic requirements on the decay leptons are applied according to the experimental acceptance. The measured \(Z\)-boson \(p_{\text {T}} \) distribution is rebinned to match the coarser bins of the \(W\)-boson \(p_{\text {T}} \) distribution, which was measured using only 30 pb\(^{-1}\) of data. The theoretical prediction is in agreement with the experimental measurements for the region with \(p_{\text {T}} <30\,\text {GeV}\), which is relevant for the measurement of the W-boson mass.
Fig. 3

a Differential Z-boson cross section as a function of boson rapidity, and b differential \(W^+\) and \(W^-\) cross sections as a function of charged decay-lepton pseudorapidity at \(\sqrt{s}=7\) \(\text {TeV}\) [41]. The measured cross sections are compared to the Powheg+Pythia 8 predictions, corrected to NNLO using DYNNLO with the CT10nnlo PDF set. The error bars show the total experimental uncertainties, including luminosity uncertainty, and the bands show the PDF uncertainties of the predictions

The predictions of RESBOS [89, 90], DYRes [91] and Powheg MiNLO+Pythia 8 [96, 97] are also considered. All predict a harder \(p_{\text {T}} ^W\) distribution for a given \(p_{\text {T}} ^Z\) distribution, compared to Pythia  8 AZ. Assuming the latter can be adjusted to match the measurement of Ref. [44], the corresponding \(p_{\text {T}} ^W\) distribution induces a discrepancy with the detector-level \(u_{\text {T}}\) and \(u_{\parallel }^\ell \) distributions observed in the W-boson data, as discussed in Sect. 11.2. This behaviour is observed using default values for the non-perturbative parameters of these programs, but is not expected to change significantly under variations of these parameters. These predictions are therefore not used in the determination of \(m_W\) or its uncertainty.

Figure 2 compares the reconstruction-level \(p_{\text {T}} ^\ell \) and \(m_{\mathrm {T}}\) distributions obtained with Powheg+Pythia 8 AZNLO, DYRes and Powheg MiNLO+Pythia 8 to those of Pythia  8 AZ.2 The effect of varying the \(p_{\text {T}} ^W\) distribution is largest at high \(p_{\text {T}} ^\ell \), which explains why the uncertainty due to the \(p_{\text {T}} ^W\) modelling is reduced when limiting the \(p_{\text {T}} ^\ell \) fitting range as described in Sect. 11.3.

6.4 Reweighting procedure

The W and Z production and decay model described above is applied to the Powheg+Pythia 8 samples through an event-by-event reweighting. Equation (3) expresses the factorisation of the cross section into the three-dimensional boson production phase space, defined by the variables m, \(p_{\text {T}} \), and y, and the two-dimensional boson decay phase space, defined by the variables \(\theta \) and \(\phi \). Accordingly, a prediction of the kinematic distributions of vector bosons and their decay products can be transformed into another prediction by applying separate reweighting of the three-dimensional boson production phase-space distributions, followed by a reweighting of the angular decay distributions.
Fig. 4

The a \(A_0\) and b \(A_2\) angular coefficients in Z-boson events as a function of \(p_{\text {T}} ^{\ell \ell }\) [42]. The measured coefficients are compared to the DYNNLO predictions using the CT10nnlo PDF set. The error bars show the total experimental uncertainties, and the bands show the uncertainties assigned to the DYNNLO predictions

The reweighting is performed in several steps. First, the inclusive rapidity distribution is reweighted according to the NNLO QCD predictions evaluated with DYNNLO. Then, at a given rapidity, the vector-boson transverse-momentum shape is reweighted to the Pythia 8 prediction with the AZ tune. This procedure provides the transverse-momentum distribution of vector bosons predicted by Pythia 8, preserving the rapidity distribution at NNLO. Finally, at given rapidity and transverse momentum, the angular variables are reweighted according to:
$$\begin{aligned}&w(\cos \theta ,\phi , p_{\text {T}},y) = \frac{1+\cos ^{2}\theta +\sum _i \, A'_i(p_{\text {T}},y) \, P_i(\cos \theta ,\phi )}{1+\cos ^{2}\theta +\sum _i \, A_i(p_{\text {T}},y) \, P_i(\cos \theta ,\phi )}, \end{aligned}$$
where \(A'_i\) are the angular coefficients evaluated at \(O(\alpha _{\mathrm s}^2)\), and \(A_i\) are the angular coefficients of the Powheg+Pythia 8 samples. This reweighting procedure neglects the small dependence of the two-dimensional (\(p_{\text {T}}\),y) distribution and of the angular coefficients on the final state invariant mass. The procedure is used to include the corrections described in Sects. 6.2 and 6.3, as well as to estimate the impact of the QCD modelling uncertainties described in Sect. 6.5.
Table 3

Systematic uncertainties in the \(m_W\) measurement due to QCD modelling, for the different kinematic distributions and W-boson charges. Except for the case of PDFs, the same uncertainties apply to \(W^+\) and \(W^-\). The fixed-order PDF uncertainty given for the separate \(W^+\) and \(W^-\) final states corresponds to the quadrature sum of the CT10nnlo uncertainty variations; the charge-combined uncertainty also contains a \(3.8\,\text {MeV}\) contribution from comparing CT10nnlo to CT14 and MMHT2014

W-boson charge

\(W^+\)

\(W^-\)

Combined

Kinematic distribution

\(p_{\text {T}} ^\ell \)

\(m_{\mathrm {T}}\)

\(p_{\text {T}} ^\ell \)

\(m_{\mathrm {T}}\)

\(p_{\text {T}} ^\ell \)

\(m_{\mathrm {T}}\)

\(\delta m_W\) [MeV]

     Fixed-order PDF uncertainty

13.1

14.9

12.0

14.2

8.0

8.7

     AZ tune

3.0

3.4

3.0

3.4

3.0

3.4

     Charm-quark mass

1.2

1.5

1.2

1.5

1.2

1.5

     Parton shower \(\mu _\text {F}\) with heavy-flavour decorrelation

5.0

6.9

5.0

6.9

5.0

6.9

     Parton shower PDF uncertainty

3.6

4.0

2.6

2.4

1.0

1.6

     Angular coefficients

5.8

5.3

5.8

5.3

5.8

5.3

     Total

15.9

18.1

14.8

17.2

11.6

12.9

The validity of the reweighting procedure is tested at particle level by generating independent W-boson samples using the CT10nnlo and NNPDF3.0 [98] NNLO PDF sets, and the same value of \(m_W\). The relevant kinematic distributions are calculated for both samples and used to reweight the CT10nnlo sample to the NNPDF3.0 one. The procedure described in Sect. 2.2 is then used to determine the value of \(m_W\) by fitting the NNPDF3.0 sample using templates from the reweighted CT10nnlo sample. The fitted value agrees with the input value within \(1.5 \pm 2.0\,\,\text {MeV}\). The statistical precision of this test is used to assign the associated systematic uncertainty.

The resulting model is tested by comparing the predicted Z-boson differential cross section as a function of rapidity, the W-boson differential cross section as a function of lepton pseudorapidity, and the angular coefficients in Z-boson events, to the corresponding ATLAS measurements [41, 42]. The comparison with the measured W and Z cross sections is shown in Fig. 3. Satisfactory agreement between the measurements and the theoretical predictions is observed. A \(\chi ^2\) compatibility test is performed for the three distributions simultaneously, including the correlations between the uncertainties. The compatibility test yields a \(\chi ^2/\)dof value of 45 / 34. Other NNLO PDF sets such as NNPDF3.0, CT14 [99], MMHT2014 [100], and ABM12 [101] are in worse agreement with these distributions. Based on the quantitative comparisons performed in Ref. [41], only CT10nnlo, CT14 and MMHT2014 are considered further. The better agreement obtained with CT10nnlo can be ascribed to the weaker suppression of the strange quark density compared to the u- and d-quark sea densities in this PDF set.

The predictions of the angular coefficients in Z-boson events are compared to the ATLAS measurement at \(\sqrt{s}=8\,\text {TeV}\) [42]. Good agreement between the measurements and DYNNLO is observed for the relevant coefficients, except for \(A_2\), where the measurement is significantly below the prediction. As an example, Fig. 4 shows the comparison for \(A_0\) and \(A_2\) as a function of \(p_{\text {T}} ^Z\). For \(A_2\), an additional source of uncertainty in the theoretical prediction is considered to account for the observed disagreement with data, as discussed in Sect. 6.5.3.

6.5 Uncertainties in the QCD modelling

Several sources of uncertainty related to the perturbative and non-perturbative modelling of the strong interaction affect the dynamics of the vector-boson production and decay [33, 102, 103, 104]. Their impact on the measurement of \(m_W\) is assessed through variations of the model parameters of the predictions for the differential cross sections as functions of the boson rapidity, transverse-momentum spectrum at a given rapidity, and angular coefficients, which correspond to the second, third, and fourth terms of the decomposition of Eq. (2), respectively. The parameter variations used to estimate the uncertainties are propagated to the simulated event samples by means of the reweighting procedure described in Sect. 6.4. Table 3 shows an overview of the uncertainties due to the QCD modelling which are discussed below.

6.5.1 Uncertainties in the fixed-order predictions

The imperfect knowledge of the PDFs affects the differential cross section as a function of boson rapidity, the angular coefficients, and the \(p_{\text {T}} ^W\) distribution. The PDF contribution to the prediction uncertainty is estimated with the CT10nnlo PDF set by using the Hessian method [105]. There are 25 error eigenvectors, and a pair of PDF variations associated with each eigenvector. Each pair corresponds to positive and negative 90% CL excursions along the corresponding eigenvector. Symmetric PDF uncertainties are defined as the mean value of the absolute positive and negative excursions corresponding to each pair of PDF variations. The overall uncertainty of the CT10nnlo PDF set is scaled to 68% CL by applying a multiplicative factor of 1/1.645.

The effect of PDF variations on the rapidity distributions and angular coefficients are evaluated with DYNNLO, while their impact on the W-boson \(p_{\text {T}} \) distribution is evaluated using Pythia 8 and by reweighting event-by-event the PDFs of the hard-scattering process, which are convolved with the LO matrix elements. Similarly to other uncertainties which affect the \(p_{\text {T}} ^W\) distribution (Sect. 6.5.2), only relative variations of the \(p_{\text {T}} ^W\) and \(p_{\text {T}} ^Z\) distributions induced by the PDFs are considered. The PDF variations are applied simultaneously to the boson rapidity, angular coefficients, and transverse-momentum distributions, and the overall PDF uncertainty is evaluated with the Hessian method as described above.

Uncertainties in the PDFs are the dominant source of physics-modelling uncertainty, contributing about 14 and \(13\,\text {MeV}\) when averaging \(p_{\text {T}} ^\ell \) and \(m_{\mathrm {T}}\) fits for \(W^+\) and \(W^-\), respectively. The PDF uncertainties are very similar when using \(p_{\text {T}} ^\ell \) or \(m_{\mathrm {T}}\) for the measurement. They are strongly anti-correlated between positively and negatively charged W bosons, and the uncertainty is reduced to \(7.4\,\text {MeV}\) on average for \(p_{\text {T}} ^\ell \) and \(m_{\mathrm {T}}\) fits, when combining opposite-charge categories. The anti-correlation of the PDF uncertainties is due to the fact that the total light-quark sea PDF is well constrained by deep inelastic scattering data, whereas the u-, d-, and s-quark decomposition of the sea is less precisely known [106]. An increase in the \(\bar{u}\) PDF is at the expense of the \(\bar{d}\) PDF, which produces opposite effects in the longitudinal polarisation of positively and negatively charged W bosons [37].

Other PDF sets are considered as alternative choices. The envelope of values of \(m_W\) extracted with the MMHT2014 and CT14 NNLO PDF sets is considered as an additional PDF uncertainty of \(3.8\,\text {MeV}\), which is added in quadrature after combining the \(W^+\) and \(W^-\) categories, leading to overall PDF uncertainties of \(8.0\,\text {MeV}\) and \(8.7\,\text {MeV}\) for \(p_{\text {T}} ^\ell \) and \(m_{\mathrm {T}}\) fits, respectively.

The effect of missing higher-order corrections on the NNLO predictions of the rapidity distributions of Z bosons, and the pseudorapidity distributions of the decay leptons of W bosons, is estimated by varying the renormalisation and factorisation scales by factors of 0.5 and 2.0 with respect to their nominal value \(\mu _\text {R} = \mu _\text {F} = m_V\) in the DYNNLO predictions. The corresponding relative uncertainty in the normalised distributions is of the order of 0.1–0.3%, and significantly smaller than the PDF uncertainties. These uncertainties are expected to have a negligible impact on the measurement of \(m_W\), and are not considered further.

The effect of the LHC beam-energy uncertainty of 0.65% [107] on the fixed-order predictions is studied. Relative variations of 0.65% around the nominal value of \(3.5\,\text {TeV}\) are considered, yielding variations of the inclusive \(W^+\) and \(W^-\) cross sections of 0.6 and 0.5%, respectively. No significant dependence as a function of lepton pseudorapidity is observed in the kinematic region used for the measurement, and the dependence as a function of \(p_{\text {T}} ^\ell \) and \(m_{\mathrm {T}}\) is expected to be even smaller. This uncertainty is not considered further.

6.5.2 Uncertainties in the parton shower predictions

Several sources of uncertainty affect the Pythia 8 parton shower model used to predict the transverse momentum of the W boson. The values of the AZ tune parameters, determined by fits to the measurement of the Z-boson transverse momentum, are affected by the experimental uncertainty of the measurement. The corresponding uncertainties are propagated to the \(p_{\text {T}} ^W\) predictions through variations of the orthogonal eigenvector components of the parameters error matrix [44]. The resulting uncertainty in \(m_W\) is \(3.0\,\text {MeV}\) for the \(p_{\text {T}} ^\ell \) distribution, and \(3.4\,\text {MeV}\) for the \(m_{\mathrm {T}}\) distribution. In the present analysis, the impact of \(p_{\text {T}} ^W\) distribution uncertainties is in general smaller when using \(p_{\text {T}} ^\ell \) than when using \(m_{\mathrm {T}}\), as a result of the comparatively narrow range used for the \(p_{\text {T}} ^\ell \) distribution fits.

Other uncertainties affecting predictions of the transverse-momentum spectrum of the W boson at a given rapidity, are propagated by considering relative variations of the \(p_{\text {T}} ^W\) and \(p_{\text {T}} ^Z\) distributions. The procedure is based on the assumption that model variations, when applied to \(p_{\text {T}} ^Z\), can be largely reabsorbed into new values of the AZ tune parameters fitted to the \(p_{\text {T}} ^Z\) data. Variations that cannot be reabsorbed by the fit are excluded, since they would lead to a significant disagreement of the prediction with the measurement of \(p_{\text {T}} ^Z\). The uncertainties due to model variations which are largely correlated between \(p_{\text {T}} ^W\) and \(p_{\text {T}} ^Z\) cancel in this procedure. In contrast, the procedure allows a correct estimation of the uncertainties due to model variations which are uncorrelated between \(p_{\text {T}} ^W\) and \(p_{\text {T}} ^Z\), and which represent the only relevant sources of theoretical uncertainties in the propagation of the QCD modelling from \(p_{\text {T}} ^Z\) to \(p_{\text {T}} ^W\).

Uncertainties due to variations of parton shower parameters that are not fitted to the \(p_{\text {T}} ^Z\) measurement include variations of the masses of the charm and bottom quarks, and variations of the factorisation scale used for the QCD ISR. The mass of the charm quark is varied in Pythia 8, conservatively, by \(\pm \,\, 0.5\,\text {GeV}\) around its nominal value of \(1.5\,\text {GeV}\). The resulting uncertainty contributes \(1.2\,\text {MeV}\) for the \(p_{\text {T}} ^\ell \) fits, and \(1.5\,\text {MeV}\) for the \(m_{\mathrm {T}}\) fits. The mass of the bottom quark is varied in Pythia 8, conservatively, by \(\pm \,\,0.8\,\text {GeV}\) around its nominal value of \(4.8\,\text {GeV}\). The resulting variations have a negligible impact on the transverse-momentum distributions of Z and W bosons, and are not considered further.

The uncertainty due to higher-order QCD corrections to the parton shower is estimated through variations of the factorisation scale, \(\mu _\text {F}\), in the QCD ISR by factors of 0.5 and 2.0 with respect to the central choice \(\mu _\text {F}^2 = p_{\text {T},0}^2 + p_{\text {T}} ^2\), where \(p_{\text {T},0}\) is an infrared cut-off, and \(p_{\text {T}}\) is the evolution variable of the parton shower [108]. Variations of the renormalisation scale in the QCD ISR are equivalent to a redefinition of \(\alpha _{\mathrm s}(m_Z)\) used for the QCD ISR, which is fixed from the fits to the \(p_{\text {T}} ^Z\) data. As a consequence, variations of the ISR renormalisation scale do not apply when estimating the uncertainty in the predicted \(p_{\text {T}} ^W\) distribution.

Higher-order QCD corrections are expected to be largely correlated between W-boson and Z-boson production induced by the light quarks, u, d, and s, in the initial state. However, a certain degree of decorrelation between W- and Z-boson transverse-momentum distributions is expected, due to the different amounts of heavy-quark-initiated production, where heavy refers to charm and bottom flavours. The physical origin of this decorrelation can be ascribed to the presence of independent QCD scales corresponding to the three-to-four flavours and four-to-five flavours matching scales \(\mu _c\) and \(\mu _b\) in the variable-flavour-number scheme PDF evolution [109], which are of the order of the charm- and bottom-quark masses, respectively. To assess this effect, the variations of \(\mu _\text {F}\) in the QCD ISR are performed simultaneously for all light-quark \(q\bar{q} \rightarrow W,Z\) processes, with \(q = u,d,s\), but independently for each of the \(c\bar{c} \rightarrow Z\), \(b\bar{b} \rightarrow Z\), and \(c\bar{q} \rightarrow W\) processes, where \(q = d,s\). The effect of the \(c\bar{q} \rightarrow W\) variations on the determination of \(m_W\) is reduced by a factor of two, to account for the presence of only one heavy-flavour quark in the initial state. The resulting uncertainty in \(m_W\) is \(5.0\,\text {MeV}\) for the \(p_{\text {T}} ^\ell \) distribution, and \(6.9\,\text {MeV}\) for the \(m_{\mathrm {T}}\) distribution. Since the \(\mu _\text {F}\) variations affect all the branchings of the shower evolution and not only vertices involving heavy quarks, this procedure is expected to yield a sufficient estimate of the \(\mu _{c,b}\)-induced decorrelation between the W- and Z-boson \(p_{\text {T}} \) distributions. Treating the \(\mu _\text {F}\) variations as correlated between all quark flavours, but uncorrelated between W- and Z-boson production, would yield a systematic uncertainty in \(m_W\) of approximately 30\(\,\text {MeV}\).

The predictions of the Pythia 8 MC generator include a reweighting of the first parton shower emission to the leading-order W+jet cross section, and do not include matching corrections to the higher-order W+jet cross section. As discussed in Sect. 11.2, predictions matched to the NLO W+jet cross section, such as Powheg MiNLO+Pythia 8 and DYRes, are in disagreement with the observed \(u^\ell _\parallel \) distribution and cannot be used to provide a reliable estimate of the associated uncertainty. The \(u^\ell _\parallel \) distribution, on the other hand, validates the Pythia 8 AZ prediction and its uncertainty, which gives confidence that missing higher-order corrections to the W-boson \(p_{\text {T}}\) distribution are small in comparison to the uncertainties that are already included, and can be neglected at the present level of precision.

The sum in quadrature of the experimental uncertainties of the AZ tune parameters, the variations of the mass of the charm quark, and the factorisation scale variations, leads to uncertainties on \(m_W\) of 6.0 and \(7.8\,\text {MeV}\) when using the \(p_{\text {T}} ^\ell \) distribution and the \(m_{\mathrm {T}}\) distribution, respectively. These sources of uncertainty are taken as fully correlated between the electron and muon channels, the positively and negatively charged W-boson production, and the \(|\eta _\ell |\) bins.

The Pythia 8 parton shower simulation employs the CTEQ6L1 leading-order PDF set. An additional independent source of PDF-induced uncertainty in the \(p_{\text {T}} ^W\) distribution is estimated by comparing several choices of the leading-order PDF used in the parton shower, corresponding to the CT14lo, MMHT2014lo and NNPDF2.3lo [110] PDF sets. The PDFs which give the largest deviation from the nominal ratio of the \(p_{\text {T}} ^W\) and \(p_{\text {T}} ^Z\) distributions are used to estimate the uncertainty. This procedure yields an uncertainty of about \(4\,\text {MeV}\) for \(W^+\), and of about \(2.5\,\text {MeV}\) for \(W^-\). Similarly to the case of fixed-order PDF uncertainties, there is a strong anti-correlation between positively and negatively charged W bosons, and the uncertainty is reduced to about \(1.5\,\text {MeV}\) when combining positive- and negative-charge categories.

The prediction of the \(p_{\text {T}} ^W\) distribution relies on the \(p_{\text {T}}\)-ordered parton shower model of the Pythia 8 MC generator. In order to assess the impact of the choice of parton shower model on the determination of \(m_W\), the Pythia 8 prediction of the ratio of the \(p_{\text {T}} ^W\) and \(p_{\text {T}} ^Z\) distributions is compared to the corresponding prediction of the Herwig 7 MC generator [111, 112], which implements an angular-ordered parton shower model. Differences between the Pythia 8 and Herwig 7 predictions are smaller than the uncertainties in the Pythia 8 prediction, and no additional uncertainty is considered.

6.5.3 Uncertainties in the angular coefficients

The full set of angular coefficients can only be measured precisely for the production of Z bosons. The accuracy of the NNLO predictions of the angular coefficients is validated by comparison to the Z-boson measurement, and extrapolated to W-boson production assuming that NNLO predictions have similar accuracy for the W- and Z-boson processes. The ATLAS measurement of the angular coefficients in Z-boson production at a centre-of-mass energy of \(\sqrt{s} = 8\,\text {TeV}\) [42] is used for this validation. The \(O(\alpha _{\mathrm s}^2)\) predictions, evaluated with DYNNLO, are in agreement with the measurements of the angular coefficients within the experimental uncertainties, except for the measurement of \(A_2\) as a function of Z-boson \(p_{\text {T}}\).

Two sources of uncertainty affecting the modelling of the angular coefficients are considered, and propagated to the W-boson predictions. One source is defined from the experimental uncertainty of the Z-boson measurement of the angular coefficients which is used to validate the NNLO predictions. The uncertainty in the corresponding W-boson predictions is estimated by propagating the experimental uncertainty of the Z-boson measurement as follows. A set of pseudodata distributions are obtained by fluctuating the angular coefficients within the experimental uncertainties, preserving the correlations between the different measurement bins for the different coefficients. For each pseudoexperiment, the differences in the \(A_i\) coefficients between fluctuated and nominal Z-boson measurement results are propagated to the corresponding coefficient in W-boson production. The corresponding uncertainty is defined from the standard deviation of the \(m_W\) values as estimated from the pseudodata distributions.

The other source of uncertainty is considered to account for the disagreement between the measurement and the NNLO QCD predictions observed for the \(A_2\) angular coefficient as a function of the Z-boson \(p_{\text {T}}\) (Fig. 4). The corresponding uncertainty in \(m_W\) is estimated by propagating the difference in \(A_2\) between the Z-boson measurement and the theoretical prediction to the corresponding coefficient in W-boson production. The corresponding uncertainty in the measurement of \(m_W\) is \(1.6\,\text {MeV}\) for the extraction from the \(p_{\text {T}} ^\ell \) distribution. Including this contribution, total uncertainties of 5.8 and \(5.3\,\text {MeV}\) due to the modelling of the angular coefficients are estimated in the determination of the W-boson mass from the \(p_{\text {T}} ^\ell \) and \(m_{\mathrm {T}}\) distributions, respectively. The uncertainty is dominated by the experimental uncertainty of the Z-boson measurement used to validate the theoretical predictions.

7 Calibration of electrons and muons

Any imperfect calibration of the detector response to electrons and muons impacts the measurement of the W-boson mass, as it affects the position and shape of the Jacobian edges reflecting the value of \(m_W\). In addition, the \(p_{\text {T}} ^\ell \) and \(m_{\mathrm {T}}\) distributions are broadened by the electron-energy and muon-momentum resolutions. Finally, the lepton-selection efficiencies depend on the lepton pseudorapidity and transverse momentum, further modifying these distributions. Corrections to the detector response are derived from the data, and presented below. In most cases, the corrections are applied to the simulation, with the exception of the muon sagitta bias corrections and electron energy response corrections, which are applied to the data. Backgrounds to the selected \(Z\rightarrow \ell \ell \) samples are taken into account using the same procedures as discussed in Sect. 9. Since the Z samples are used separately for momentum calibration and efficiency measurements, as well as for the recoil response corrections discussed in Sect. 8, correlations among the corresponding uncertainties can appear. These correlations were investigated and found to be negligible.

7.1 Muon momentum calibration

As described in Sect. 5.1, the kinematic parameters of selected muons are determined from the associated inner-detector tracks. The accuracy of the momentum measurement is limited by imperfect knowledge of the detector alignment and resolution, of the magnetic field, and of the amount of passive material in the detector.

Biases in the reconstructed muon track momenta are classified as radial or sagitta biases. The former originate from detector movements along the particle trajectory and can be corrected by an \(\eta \)-dependent, charge-independent momentum-scale correction. The latter typically originate from curl distortions or linear twists of the detector around the z-axis [113], and can be corrected with \(\eta \)-dependent correction factors proportional to \(q\times p_{\text {T}} ^\ell \), where q is the charge of the muon. The momentum scale and resolution corrections are applied to the simulation, while the sagitta bias correction is applied to the data:
$$\begin{aligned}&p_{\text {T}} ^{\text {MC,corr}} = p_{\text {T}} ^{\text {MC}} \times \left[ 1 + \alpha (\eta ,\phi )\right] \\&\qquad \qquad \qquad \qquad \times \left[ 1 + \beta _{\text {curv}}(\eta ) \cdot G(0,1) \cdot p_{\text {T}}^{\text {MC}}\right] ,\\ \nonumber&p_{\text {T}} ^{\text {data,corr}} = \frac{p_{\text {T}} ^{\text {data}}}{1 + q \cdot \delta (\eta ,\phi ) \cdot p_{\text {T}} ^{\text {data}}}, \end{aligned}$$
where \(p_{\text {T}} ^{\text {data,MC}}\) is the uncorrected muon transverse momentum in data and simulation, G(0, 1) are normally distributed random variables with mean zero and unit width, and \(\alpha \), \(\beta _{\text {curv}}\), and \(\delta \) represent the momentum scale, intrinsic resolution and sagitta bias corrections, respectively. Multiple-scattering contributions to the resolution are relevant at low \(p_{\text {T}}\), and the corresponding corrections are neglected.

Momentum scale and resolution corrections are derived using \(Z\rightarrow \mu \mu \) decays, following the method described in Ref. [40]. Template histograms of the dimuon invariant mass are constructed from the simulated event samples, including momentum scale and resolution corrections in narrow steps within a range covering the expected uncertainty. The optimal values of \(\alpha \) and \(\beta _{\mathrm {curv}}\) are determined by means of a \(\chi ^2\) minimisation, comparing data and simulation in the range of twice the standard deviation on each side of the mean value of the invariant mass distribution. In the first step, the corrections are derived by averaging over \(\phi \), and for 24 pseudorapidity bins in the range \(-\,2.4< \eta _\ell < 2.4\). In the second iteration, \(\phi \)-dependent correction factors are evaluated in coarser bins of \(\eta _\ell \). The typical size of \(\alpha \) varies from − 0.0005 to − 0.0015 depending on \(\eta _\ell \), while \(\beta _{\text {curv}}\) values increase from \(0.2\, \text {TeV}^{-1}\) in the barrel to \(0.6\, \text {TeV}^{-1}\) in the high \(\eta _\ell \) region. Before the correction, the \(\phi \)-dependence has an amplitude at the level of 0.1%.

The \(\alpha \) and \(\beta _{\mathrm {curv}}\) corrections are sensitive to the following aspects of the calibration procedure, which are considered for the systematic uncertainty: the choice of the fitting range, methodological biases, background contributions, theoretical modelling of Z-boson production, non-linearity of the corrections, and material distribution in the ID. The uncertainty due to the choice of fitting range is estimated by varying the range by \({\pm }\,10\%\), and repeating the procedure. The uncertainty due to the fit methodology is estimated by comparing the template fit results with an alternative approach, based on an iterative \(\chi ^2\) minimisation. Background contributions from gauge-boson pair and top-quark pair production are estimated using the simulation. The uncertainty in these background contributions is evaluated by varying their normalisation within the theoretical uncertainties on the production cross sections. The uncertainty in the theoretical modelling of Z-boson production is evaluated by propagating the effect of electroweak corrections to QED FSR, QED radiation of fermion pairs, and other NLO electroweak corrections described in Sect. 6.1. The experimental uncertainty in the value of the Z-boson mass used as input is also accounted for. These sources of uncertainty are summed in quadrature, yielding an uncertainty \(\delta \alpha \) in the muon momentum scale correction of approximately \(0.5 \times 10^{-4}\); these sources are considered fully correlated across muon pseudorapidity.

The systematic uncertainty in the muon momentum scale due to the extrapolation from the \(Z\rightarrow \mu \mu \) momentum range to the \(W\rightarrow \mu \nu \) momentum range is estimated by evaluating momentum-scale corrections as a function of \(1/p_{\text {T}} \) for muons in various \(|\eta |\) ranges. The extrapolation uncertainty \(\delta \alpha \) is parameterised as follows:where Open image in new window is the average \(p_{\text {T}}\) of muons in W-boson events, and \(p_0\) and \(p_1\) are free parameters. If the momentum-scale corrections are independent of \(1/p_{\text {T}} \), the fitting parameters are expected to be \(p_0=1\) and \(p_1=0\). Deviations of \(p_1\) from zero indicate a possible momentum dependence. The fitted values of \(\delta \alpha \) are shown in Fig. 5a, and are consistent with one, within two standard deviations of the statistical error. The corresponding systematic uncertainty in \(m_W\) is defined assuming, in each bin of \(|\eta |\), a momentum non-linearity given by the larger of the fitted value of \(p_1\) and its uncertainty. This source of uncertainty is considered uncorrelated across muon pseudorapidity given that \(p_1\) is dominated by statistical fluctuations. The effect of the imperfect knowledge of the material in the ID is studied using simulated event samples including an increase of the ID material by 10%, according to the uncertainty estimated in Ref. [114]. The impact of this variation is found to be negligible in comparison with the uncertainties discussed above.
Fig. 5

a Residual muon momentum scale corrections as a function of muon \(1/p_{\text {T}}\) in four pseudorapidity regions, obtained with \(Z\rightarrow \mu \mu \) events. The points are fitted using a linear function which parameterises the extrapolation of the muon momentum scale correction from Z to W events, as explained in the text. The error bars on the points show statistical uncertainties only. b Sagitta bias, \(\delta \), as a function of \(\eta _\ell \) averaged over \(\phi _\ell \). The results are obtained with the \(Z\rightarrow \mu \mu \) and E / p methods and the combination of the two. The results obtained with the \(Z\rightarrow \mu \mu \) method are corrected for the global sagitta bias. The E / p method uses electrons from \(W\rightarrow e\nu \) decays. The two measurements are combined assuming they are uncorrelated. The error bars on the points show statistical uncertainties only

Two methods are used for the determination of the sagitta bias \(\delta \). The first method exploits \(Z \rightarrow \mu \mu \) events. Muons are categorised according to their charge and pseudorapidity, and for each of these categories, the position of the peak in the dimuon invariant mass distribution is determined for data and simulation. The procedure allows the determination of the charge dependence of the momentum scale for \(p_{\mathrm T}\) values of approximately \(42\,\text {GeV}\), which corresponds to the average transverse momentum of muons from Z-boson decays. The second method exploits identified electrons in a sample of \(W\rightarrow e\nu \) decays. It is based on the ratio of the measured electron energy deposited in the calorimeter, E, to the electron momentum, p, measured in the ID. A clean sample of \(W\rightarrow e\nu \) events with tightly identified electrons [38] is selected. Assuming that the response of the electromagnetic calorimeter is independent of the charge of the incoming particle, charge-dependent ID track momentum biases are extracted from the average differences in E / p for electrons and positrons [113]. This method benefits from a larger event sample compared to the first method, and allows the determination of charge-dependent corrections for \(p_{\mathrm T}\) values of approximately \(38\,\text {GeV}\), which corresponds to the average transverse momentum of muons in W-boson decays. The sagitta bias correction factors are derived using both methods separately in 40 \(\eta \) bins and 40 \(\phi \) bins. The results are found to agree within uncertainties and are combined, as illustrated in Fig. 5b. The combined correction uncertainty is dominated by the finite size of the event samples.

Figure 6 shows the dimuon invariant mass distribution of \(Z \rightarrow \mu \mu \) decays in data and simulation, after applying all corrections. Table 4 summarises the effect of the muon momentum scale and resolution uncertainties on the determination of \(m_W\). The dominant systematic uncertainty in the momentum scale is due to the extrapolation of the correction from the Z-boson momentum range to the W-boson momentum range. The extrapolation uncertainty \(\delta \alpha \) is (2–\(5)\times 10^{-5}\) for \(|\eta _\ell |<2.0\), and (4–\(7)\times 10^{-4}\) for \(|\eta _\ell |>2.0\). Systematic uncertainties from other sources are relatively small. The systematic uncertainty of the resolution corrections is dominated by the statistical uncertainty of the Z-boson event sample, and includes a contribution from the imperfect closure of the method. The latter is defined from the residual difference between the standard deviations of the dimuon invariant mass in data and simulation, after applying resolution corrections.
Fig. 6

Dimuon invariant mass distribution in \(Z\rightarrow \mu \mu \) events. The data are compared to the simulation including signal and background contributions. Corrections for momentum scale and resolution, and for reconstruction, isolation, and trigger efficiencies are applied to the muons in the simulated events. Background events contribute less than 0.2% of the observed distribution. The lower panel shows the data-to-prediction ratio, with the error bars showing the statistical uncertainty

7.2 Muon selection efficiency

The selection of muon candidates in \(W\rightarrow \mu \nu \) and \(Z\rightarrow \mu \mu \) events requires an isolated track reconstructed in the inner detector and in the muon spectrometer. In addition, the events are required to pass the muon trigger selection. Differences in the efficiency of the reconstruction and selection requirements between data and simulation can introduce a systematic shift in the measurement of the W-boson mass, and have to be corrected. In particular, the extraction of \(m_W\) is sensitive to the dependence of the trigger, reconstruction and isolation efficiencies on the muon \(p_{\text {T}}\) and on the projection of the recoil on the lepton transverse momentum, \(u^\ell _\parallel \).

For muons with \(p_{\text {T}}\) larger than approximately \(15\,\text {GeV}\) the detector simulation predicts constant efficiency as a function of \(p_{\text {T}} ^\ell \), both for the muon trigger selection and the track reconstruction. In contrast, the efficiency of the isolation requirement is expected to vary as a function of \(p_{\text {T}} ^\ell \) and \(u^\ell _\parallel \). The efficiency corrections also affect the muon selection inefficiency, and hence the estimation of the \(Z\rightarrow \mu \mu \) background, which contributes to the \(W\rightarrow \mu \nu \) selection when one of the decay muons fails the muon reconstruction or kinematic selection requirements.
Table 4

Systematic uncertainties in the \(m_W\) measurement from muon calibration and efficiency corrections, for the different kinematic distributions and \(|\eta _\ell |\) categories, averaged over lepton charge. The momentum-scale uncertainties include the effects of both the momentum scale and linearity corrections. Combined uncertainties are evaluated as described in Sect. 2.2

\(|\eta _\ell |\) range

[0.0, 0.8]

[0.8, 1.4]

[1.4, 2.0]

[2.0, 2.4]

Combined

Kinematic distribution

\(p_{\text {T}} ^\ell \)

\(m_{\mathrm {T}}\)

\(p_{\text {T}} ^\ell \)

\(m_{\mathrm {T}}\)

\(p_{\text {T}} ^\ell \)

\(m_{\mathrm {T}}\)

\(p_{\text {T}} ^\ell \)

\(m_{\mathrm {T}}\)

\(p_{\text {T}} ^\ell \)

\(m_{\mathrm {T}}\)

\(\delta m_W\) [MeV]

     Momentum scale

8.9

9.3

14.2

15.6

27.4

29.2

111.0

115.4

8.4

8.8

     Momentum resolution

1.8

2.0

1.9

1.7

1.5

2.2

3.4

3.8

1.0

1.2

     Sagitta bias

0.7

0.8

1.7

1.7

3.1

3.1

4.5

4.3

0.6

0.6

     Reconstruction and isolation efficiencies

4.0

3.6

5.1

3.7

4.7

3.5

6.4

5.5

2.7

2.2

     Trigger efficiency

5.6

5.0

7.1

5.0

11.8

9.1

12.1

9.9

4.1

3.2

     Total

11.4

11.4

16.9

17.0

30.4

31.0

112.0

116.1

9.8

9.7

Fig. 7

a Scale factors for the muon reconstruction, trigger and isolation efficiency obtained with the tag and probe method as a function of the muon \(p_{\mathrm T}\). Scale factors for the trigger efficiency are averaged over two data-taking periods as explained in the text. The error bars on the points show statistical uncertainties only. b Distribution of the reconstructed muons \(\eta \) in \(Z \rightarrow \mu \mu \) events. The data are compared to the simulation including signal and background contributions. Corrections for momentum scale and resolution, and for reconstruction, isolation, and trigger efficiencies are applied to the muons in the simulated events. Background events contribute less than 0.2% of the observed distribution. The lower panel shows the data-to-prediction ratio, with the error bars showing the statistical uncertainty