EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)

 CERN-PH-EP-2012-099 LHCb-PAPER-2012-008 May 24, 2012

Inclusive $W$ and $Z$ production in the forward region at
$\sqrt{s}=7$$\mathrm{\,Te\kern-2.07413ptV}$

The LHCb collaboration †††Authors are listed on the following pages.

Measurements of inclusive $W$ and $Z$ boson production cross-sections in $pp$
collisions at $\sqrt{s}=7$$\mathrm{\,Te\kern-1.00006ptV}$ using final states
containing muons are presented. The data sample corresponds to an integrated
luminosity of $37$$\mbox{\,pb}^{-1}$ collected with the LHCb detector. The $W$
and $Z$ bosons are reconstructed from muons with a transverse momentum above
20${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ and pseudorapidity between $2.0$ and
$4.5$, and, in the case of the $Z$ cross-section, a dimuon invariant mass
between $60$ and $120$${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$. The cross-
sections are measured to be $831\pm 9\pm 27\pm 29$$\rm\,pb$for $W^{+}$,
$656\pm 8\pm 19\pm 23$$\rm\,pb$for $W^{-}$ and $76.7\pm 1.7\pm 3.3\pm
2.7$$\rm\,pb$ for $Z$, where the first uncertainty is statistical, the second
is systematic and the third is due to the luminosity. Differential cross-
sections, $W$ and $Z$ cross-section ratios and the lepton charge asymmetry are
also measured in the same kinematic region. The ratios are determined to be
$\sigma_{W^{+}\rightarrow\mu^{+}\nu}/\sigma_{W^{-}\rightarrow\mu^{-}\bar{\nu}}=$$1.27\pm
0.02\pm 0.01$ and
$(\sigma_{W^{+}\rightarrow\mu^{+}\nu}+\sigma_{W^{-}\rightarrow\mu^{-}\bar{\nu}})/\sigma_{Z\rightarrow\mu\mu}=$$19.4\pm
0.5\pm 0.9$. The results are in general agreement with theoretical
predictions, performed at next-to-next-to-leading order in QCD using recently
calculated parton distribution functions.

Published in JHEP Vol. 2012, Number 6 (2012), 58, DOI: 10.1007/JHEP06(2012)

LHCb collaboration

R. Aaij38, C. Abellan Beteta33,n, A. Adametz11, B. Adeva34, M. Adinolfi43, C.
Adrover6, A. Affolder49, Z. Ajaltouni5, J. Albrecht35, F. Alessio35, M.
Alexander48, S. Ali38, G. Alkhazov27, P. Alvarez Cartelle34, A.A. Alves Jr22,
S. Amato2, Y. Amhis36, J. Anderson37, R.B. Appleby51, O. Aquines Gutierrez10,
F. Archilli18,35, A. Artamonov 32, M. Artuso53,35, E. Aslanides6, G.
Auriemma22,m, S. Bachmann11, J.J. Back45, V. Balagura28,35, W. Baldini16, R.J.
Barlow51, C. Barschel35, S. Barsuk7, W. Barter44, A. Bates48, C. Bauer10, Th.
Bauer38, A. Bay36, J. Beddow48, I. Bediaga1, S. Belogurov28, K. Belous32, I.
Belyaev28, E. Ben-Haim8, M. Benayoun8, G. Bencivenni18, S. Benson47, J.
Benton43, R. Bernet37, M.-O. Bettler17, M. van Beuzekom38, A. Bien11, S.
Bifani12, T. Bird51, A. Bizzeti17,h, P.M. Bjørnstad51, T. Blake35, F. Blanc36,
C. Blanks50, J. Blouw11, S. Blusk53, A. Bobrov31, V. Bocci22, A. Bondar31, N.
Bondar27, W. Bonivento15, S. Borghi48,51, A. Borgia53, T.J.V. Bowcock49, C.
Bozzi16, T. Brambach9, J. van den Brand39, J. Bressieux36, D. Brett51, M.
Britsch10, T. Britton53, N.H. Brook43, H. Brown49, A. Büchler-Germann37, I.
Burducea26, A. Bursche37, J. Buytaert35, S. Cadeddu15, O. Callot7, M.
Calvi20,j, M. Calvo Gomez33,n, A. Camboni33, P. Campana18,35, A. Carbone14, G.
Carboni21,k, R. Cardinale19,i,35, A. Cardini15, L. Carson50, K. Carvalho
Akiba2, G. Casse49, M. Cattaneo35, Ch. Cauet9, M. Charles52, Ph.
Charpentier35, N. Chiapolini37, M. Chrzaszcz 23, K. Ciba35, X. Cid Vidal34, G.
Ciezarek50, P.E.L. Clarke47, M. Clemencic35, H.V. Cliff44, J. Closier35, C.
Coca26, V. Coco38, J. Cogan6, E. Cogneras5, P. Collins35, A. Comerma-
Montells33, A. Contu52, A. Cook43, M. Coombes43, G. Corti35, B. Couturier35,
G.A. Cowan36, R. Currie47, C. D’Ambrosio35, P. David8, P.N.Y. David38, I. De
Bonis4, K. De Bruyn38, S. De Capua21,k, M. De Cian37, J.M. De Miranda1, L. De
Paula2, P. De Simone18, D. Decamp4, M. Deckenhoff9, H. Degaudenzi36,35, L. Del
Buono8, C. Deplano15, D. Derkach14,35, O. Deschamps5, F. Dettori39, J.
Dickens44, H. Dijkstra35, P. Diniz Batista1, F. Domingo Bonal33,n, S.
Donleavy49, F. Dordei11, A. Dosil Suárez34, D. Dossett45, A. Dovbnya40, F.
Dupertuis36, R. Dzhelyadin32, A. Dziurda23, A. Dzyuba27, S. Easo46, U.
Egede50, V. Egorychev28, S. Eidelman31, D. van Eijk38, F. Eisele11, S.
Eisenhardt47, R. Ekelhof9, L. Eklund48, Ch. Elsasser37, D. Elsby42, D.
Esperante Pereira34, A. Falabella16,e,14, C. Färber11, G. Fardell47, C.
Farinelli38, S. Farry12, V. Fave36, V. Fernandez Albor34, M. Ferro-Luzzi35, S.
Filippov30, C. Fitzpatrick47, M. Fontana10, F. Fontanelli19,i, R. Forty35, O.
Francisco2, M. Frank35, C. Frei35, M. Frosini17,f, S. Furcas20, A. Gallas
Torreira34, D. Galli14,c, M. Gandelman2, P. Gandini52, Y. Gao3, J-C.
Garnier35, J. Garofoli53, J. Garra Tico44, L. Garrido33, D. Gascon33, C.
Gaspar35, R. Gauld52, N. Gauvin36, M. Gersabeck35, T. Gershon45,35, Ph. Ghez4,
V. Gibson44, V.V. Gligorov35, C. Göbel54, D. Golubkov28, A. Golutvin50,28,35,
A. Gomes2, H. Gordon52, M. Grabalosa Gándara33, R. Graciani Diaz33, L.A.
Granado Cardoso35, E. Graugés33, G. Graziani17, A. Grecu26, E. Greening52, S.
Gregson44, O. Grünberg55, B. Gui53, E. Gushchin30, Yu. Guz32, T. Gys35, C.
Hadjivasiliou53, G. Haefeli36, C. Haen35, S.C. Haines44, T. Hampson43, S.
Hansmann-Menzemer11, N. Harnew52, J. Harrison51, P.F. Harrison45, T.
Hartmann55, J. He7, V. Heijne38, K. Hennessy49, P. Henrard5, J.A. Hernando
Morata34, E. van Herwijnen35, E. Hicks49, K. Holubyev11, P. Hopchev4, W.
Hulsbergen38, P. Hunt52, T. Huse49, R.S. Huston12, D. Hutchcroft49, D.
Hynds48, V. Iakovenko41, P. Ilten12, J. Imong43, R. Jacobsson35, A. Jaeger11,
M. Jahjah Hussein5, E. Jans38, F. Jansen38, P. Jaton36, B. Jean-Marie7, F.
Jing3, M. John52, D. Johnson52, C.R. Jones44, B. Jost35, M. Kaballo9, S.
Kandybei40, M. Karacson35, T.M. Karbach9, J. Keaveney12, I.R. Kenyon42, U.
Kerzel35, T. Ketel39, A. Keune36, B. Khanji6, Y.M. Kim47, M. Knecht36, I.
Komarov29, R.F. Koopman39, P. Koppenburg38, M. Korolev29, A. Kozlinskiy38, L.
Kravchuk30, K. Kreplin11, M. Kreps45, G. Krocker11, P. Krokovny31, F. Kruse9,
K. Kruzelecki35, M. Kucharczyk20,23,35,j, V. Kudryavtsev31, T.
Kvaratskheliya28,35, V.N. La Thi36, D. Lacarrere35, G. Lafferty51, A. Lai15,
D. Lambert47, R.W. Lambert39, E. Lanciotti35, G. Lanfranchi18, C.
Langenbruch35, T. Latham45, C. Lazzeroni42, R. Le Gac6, J. van Leerdam38,
J.-P. Lees4, R. Lefèvre5, A. Leflat29,35, J. Lefrançois7, O. Leroy6, T.
Lesiak23, L. Li3, Y. Li3, L. Li Gioi5, M. Lieng9, M. Liles49, R. Lindner35, C.
Linn11, B. Liu3, G. Liu35, J. von Loeben20, J.H. Lopes2, E. Lopez Asamar33, N.
Lopez-March36, H. Lu3, J. Luisier36, A. Mac Raighne48, F. Machefert7, I.V.
Machikhiliyan4,28, F. Maciuc10, O. Maev27,35, J. Magnin1, S. Malde52, R.M.D.
Mamunur35, G. Manca15,d, G. Mancinelli6, N. Mangiafave44, U. Marconi14, R.
Märki36, J. Marks11, G. Martellotti22, A. Martens8, L. Martin52, A. Martín
Sánchez7, M. Martinelli38, D. Martinez Santos35, A. Massafferri1, Z. Mathe12,
C. Matteuzzi20, M. Matveev27, E. Maurice6, B. Maynard53, A. Mazurov16,30,35,
G. McGregor51, R. McNulty12, M. Meissner11, M. Merk38, J. Merkel9, S.
Miglioranzi35, D.A. Milanes13, M.-N. Minard4, J. Molina Rodriguez54, S.
Monteil5, D. Moran12, P. Morawski23, R. Mountain53, I. Mous38, F. Muheim47, K.
Müller37, R. Muresan26, B. Muryn24, B. Muster36, J. Mylroie-Smith49, P.
Naik43, T. Nakada36, R. Nandakumar46, I. Nasteva1, M. Needham47, N. Neufeld35,
A.D. Nguyen36, C. Nguyen-Mau36,o, M. Nicol7, V. Niess5, N. Nikitin29, T.
Nikodem11, A. Nomerotski52,35, A. Novoselov32, A. Oblakowska-Mucha24, V.
Obraztsov32, S. Oggero38, S. Ogilvy48, O. Okhrimenko41, R. Oldeman15,d,35, M.
Orlandea26, J.M. Otalora Goicochea2, P. Owen50, B.K. Pal53, J. Palacios37, A.
Palano13,b, M. Palutan18, J. Panman35, A. Papanestis46, M. Pappagallo48, C.
Parkes51, C.J. Parkinson50, G. Passaleva17, G.D. Patel49, M. Patel50, S.K.
Paterson50, G.N. Patrick46, C. Patrignani19,i, C. Pavel-Nicorescu26, A. Pazos
Alvarez34, A. Pellegrino38, G. Penso22,l, M. Pepe Altarelli35, S.
Perazzini14,c, D.L. Perego20,j, E. Perez Trigo34, A. Pérez-Calero Yzquierdo33,
P. Perret5, M. Perrin-Terrin6, G. Pessina20, A. Petrolini19,i, A. Phan53, E.
Picatoste Olloqui33, B. Pie Valls33, B. Pietrzyk4, T. Pilař45, D. Pinci22, R.
Plackett48, S. Playfer47, M. Plo Casasus34, G. Polok23, A. Poluektov45,31, E.
Polycarpo2, D. Popov10, B. Popovici26, C. Potterat33, A. Powell52, J.
Prisciandaro36, V. Pugatch41, A. Puig Navarro33, W. Qian53, J.H. Rademacker43,
B. Rakotomiaramanana36, M.S. Rangel2, I. Raniuk40, G. Raven39, S. Redford52,
M.M. Reid45, A.C. dos Reis1, S. Ricciardi46, A. Richards50, K. Rinnert49, D.A.
Roa Romero5, P. Robbe7, E. Rodrigues48,51, F. Rodrigues2, P. Rodriguez
Perez34, G.J. Rogers44, S. Roiser35, V. Romanovsky32, M. Rosello33,n, J.
Rouvinet36, T. Ruf35, H. Ruiz33, G. Sabatino21,k, J.J. Saborido Silva34, N.
Sagidova27, P. Sail48, B. Saitta15,d, C. Salzmann37, M. Sannino19,i, R.
Santacesaria22, C. Santamarina Rios34, R. Santinelli35, E. Santovetti21,k, M.
Sapunov6, A. Sarti18,l, C. Satriano22,m, A. Satta21, M. Savrie16,e, D.
Savrina28, P. Schaack50, M. Schiller39, H. Schindler35, S. Schleich9, M.
Schlupp9, M. Schmelling10, B. Schmidt35, O. Schneider36, A. Schopper35, M.-H.
Schune7, R. Schwemmer35, B. Sciascia18, A. Sciubba18,l, M. Seco34, A.
Semennikov28, K. Senderowska24, I. Sepp50, N. Serra37, J. Serrano6, P.
Seyfert11, M. Shapkin32, I. Shapoval40,35, P. Shatalov28, Y. Shcheglov27, T.
Shears49, L. Shekhtman31, O. Shevchenko40, V. Shevchenko28, A. Shires50, R.
Silva Coutinho45, T. Skwarnicki53, N.A. Smith49, E. Smith52,46, M. Smith51, K.
Sobczak5, F.J.P. Soler48, A. Solomin43, F. Soomro18,35, B. Souza De Paula2, B.
Spaan9, A. Sparkes47, P. Spradlin48, F. Stagni35, S. Stahl11, O. Steinkamp37,
S. Stoica26, S. Stone53,35, B. Storaci38, M. Straticiuc26, U. Straumann37,
V.K. Subbiah35, S. Swientek9, M. Szczekowski25, P. Szczypka36, T. Szumlak24,
S. T’Jampens4, E. Teodorescu26, F. Teubert35, C. Thomas52, E. Thomas35, J. van
Tilburg11, V. Tisserand4, M. Tobin37, S. Tolk39, S. Topp-Joergensen52, N.
Torr52, E. Tournefier4,50, S. Tourneur36, M.T. Tran36, A. Tsaregorodtsev6, N.
Tuning38, M. Ubeda Garcia35, A. Ukleja25, U. Uwer11, V. Vagnoni14, G.
Valenti14, R. Vazquez Gomez33, P. Vazquez Regueiro34, S. Vecchi16, J.J.
Velthuis43, M. Veltri17,g, B. Viaud7, I. Videau7, D. Vieira2, X. Vilasis-
Cardona33,n, J. Visniakov34, A. Vollhardt37, D. Volyanskyy10, D. Voong43, A.
Vorobyev27, V. Vorobyev31, C. Voß55, H. Voss10, R. Waldi55, R. Wallace12, S.
Wandernoth11, J. Wang53, D.R. Ward44, N.K. Watson42, A.D. Webber51, D.
Websdale50, M. Whitehead45, J. Wicht35, D. Wiedner11, L. Wiggers38, G.
Wilkinson52, M.P. Williams45,46, M. Williams50, F.F. Wilson46, J. Wishahi9, M.
Witek23, W. Witzeling35, S.A. Wotton44, S. Wright44, S. Wu3, K. Wyllie35, Y.
Xie47, F. Xing52, Z. Xing53, Z. Yang3, R. Young47, X. Yuan3, O. Yushchenko32,
M. Zangoli14, M. Zavertyaev10,a, F. Zhang3, L. Zhang53, W.C. Zhang12, Y.
Zhang3, A. Zhelezov11, L. Zhong3, A. Zvyagin35.

1Centro Brasileiro de Pesquisas Físicas (CBPF), Rio de Janeiro, Brazil
2Universidade Federal do Rio de Janeiro (UFRJ), Rio de Janeiro, Brazil
3Center for High Energy Physics, Tsinghua University, Beijing, China
4LAPP, Université de Savoie, CNRS/IN2P3, Annecy-Le-Vieux, France
5Clermont Université, Université Blaise Pascal, CNRS/IN2P3, LPC, Clermont-
Ferrand, France
6CPPM, Aix-Marseille Université, CNRS/IN2P3, Marseille, France
7LAL, Université Paris-Sud, CNRS/IN2P3, Orsay, France
8LPNHE, Université Pierre et Marie Curie, Université Paris Diderot,
CNRS/IN2P3, Paris, France
9Fakultät Physik, Technische Universität Dortmund, Dortmund, Germany
10Max-Planck-Institut für Kernphysik (MPIK), Heidelberg, Germany
11Physikalisches Institut, Ruprecht-Karls-Universität Heidelberg, Heidelberg,
Germany
12School of Physics, University College Dublin, Dublin, Ireland
13Sezione INFN di Bari, Bari, Italy
14Sezione INFN di Bologna, Bologna, Italy
15Sezione INFN di Cagliari, Cagliari, Italy
16Sezione INFN di Ferrara, Ferrara, Italy
17Sezione INFN di Firenze, Firenze, Italy
18Laboratori Nazionali dell’INFN di Frascati, Frascati, Italy
19Sezione INFN di Genova, Genova, Italy
20Sezione INFN di Milano Bicocca, Milano, Italy
21Sezione INFN di Roma Tor Vergata, Roma, Italy
22Sezione INFN di Roma La Sapienza, Roma, Italy
23Henryk Niewodniczanski Institute of Nuclear Physics Polish Academy of
Sciences, Kraków, Poland
24AGH University of Science and Technology, Kraków, Poland
25Soltan Institute for Nuclear Studies, Warsaw, Poland
26Horia Hulubei National Institute of Physics and Nuclear Engineering,
Bucharest-Magurele, Romania
27Petersburg Nuclear Physics Institute (PNPI), Gatchina, Russia
28Institute of Theoretical and Experimental Physics (ITEP), Moscow, Russia
29Institute of Nuclear Physics, Moscow State University (SINP MSU), Moscow,
Russia
30Institute for Nuclear Research of the Russian Academy of Sciences (INR RAN),
Moscow, Russia
31Budker Institute of Nuclear Physics (SB RAS) and Novosibirsk State
University, Novosibirsk, Russia
32Institute for High Energy Physics (IHEP), Protvino, Russia
33Universitat de Barcelona, Barcelona, Spain
34Universidad de Santiago de Compostela, Santiago de Compostela, Spain
35European Organization for Nuclear Research (CERN), Geneva, Switzerland
36Ecole Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland
37Physik-Institut, Universität Zürich, Zürich, Switzerland
38Nikhef National Institute for Subatomic Physics, Amsterdam, The Netherlands
39Nikhef National Institute for Subatomic Physics and VU University Amsterdam,
Amsterdam, The Netherlands
40NSC Kharkiv Institute of Physics and Technology (NSC KIPT), Kharkiv, Ukraine
41Institute for Nuclear Research of the National Academy of Sciences (KINR),
Kyiv, Ukraine
42University of Birmingham, Birmingham, United Kingdom
43H.H. Wills Physics Laboratory, University of Bristol, Bristol, United
Kingdom
44Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom
45Department of Physics, University of Warwick, Coventry, United Kingdom
46STFC Rutherford Appleton Laboratory, Didcot, United Kingdom
47School of Physics and Astronomy, University of Edinburgh, Edinburgh, United
Kingdom
48School of Physics and Astronomy, University of Glasgow, Glasgow, United
Kingdom
49Oliver Lodge Laboratory, University of Liverpool, Liverpool, United Kingdom
50Imperial College London, London, United Kingdom
51School of Physics and Astronomy, University of Manchester, Manchester,
United Kingdom
52Department of Physics, University of Oxford, Oxford, United Kingdom
53Syracuse University, Syracuse, NY, United States
54Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), Rio de
Janeiro, Brazil, associated to 2
55Institut für Physik, Universität Rostock, Rostock, Germany, associated to 11

aP.N. Lebedev Physical Institute, Russian Academy of Science (LPI RAS),
Moscow, Russia
bUniversità di Bari, Bari, Italy
cUniversità di Bologna, Bologna, Italy
dUniversità di Cagliari, Cagliari, Italy
eUniversità di Ferrara, Ferrara, Italy
fUniversità di Firenze, Firenze, Italy
gUniversità di Urbino, Urbino, Italy
hUniversità di Modena e Reggio Emilia, Modena, Italy
iUniversità di Genova, Genova, Italy
jUniversità di Milano Bicocca, Milano, Italy
kUniversità di Roma Tor Vergata, Roma, Italy
lUniversità di Roma La Sapienza, Roma, Italy
mUniversità della Basilicata, Potenza, Italy
nLIFAELS, La Salle, Universitat Ramon Llull, Barcelona, Spain
oHanoi University of Science, Hanoi, Viet Nam

##  1 Introduction

The measurement of the production cross-sections for $W$ and $Z$ bosons
constitutes an important test of the Standard Model and provides valuable
input to constrain the proton parton density functions (PDFs). Theoretical
predictions are known to next-to-next-to-leading-order (NNLO) in perturbative
quantum chromodynamics (pQCD). These calculations are in good agreement with
recent measurements at the LHC from the ATLAS [1, 2], and the CMS [3, 4]
experiments as well as with the results from the $p\bar{p}$ collider
experiments at the $\mathrm{S}\bar{\mathrm{p}\mathrm{pS}}$ [5, 6] and the
Tevatron [7, 8, 9, 10]. The dominant theoretical uncertainty on the cross-
sections arises from the present knowledge of the PDFs and the strong coupling
constant. The accuracy strongly depends on the pseudorapidity111The
pseudorapidity $\eta$ is defined to be $\eta=-\ln\tan(\theta/2)$, where the
polar angle $\theta$ is measured with respect to the beam axis. range;
consequently, measurements by LHCb, which is fully instrumented in the forward
region $2.0<\eta<5.0$, can provide input to constrain the PDFs, both for
pseudorapidities $\eta>2.5$ and in the region which is common to ATLAS and
CMS, $2.0<\eta<2.5$. Besides the determination of the $W$ and $Z$ boson cross-
sections, the measurement of their ratios
$R_{WZ}=(\sigma_{W^{+}\rightarrow\mu^{+}\nu}+\sigma_{W^{-}\rightarrow\mu^{-}\bar{\nu}})/\sigma_{Z\rightarrow\mu\mu}$
and
$R_{W}=\sigma_{W^{+}\rightarrow\mu^{+}\nu}/\sigma_{W^{-}\rightarrow\mu^{-}\bar{\nu}}$
and of the $W$ production charge asymmetry constitute important tests of the
Standard Model, as experimental and theoretical uncertainties partially
cancel. The $W$ charge asymmetry is sensitive to the valence quark
distribution in the proton [11] and provides complementary information to the
results from deep-inelastic scattering cross-sections at HERA [12] as those
data do not strongly constrain the ratio of $u$ over $d$ quarks at low Bjorken
$x$, where $x$ is the proton momentum fraction carried by the quark.
Measurements of $W$ and $Z$ boson production at LHCb have a sensitivity to
values of $x$ as low as $1.7\times 10^{-4}$ and will contribute significantly
to the understanding of PDFs at low $x$ and reasonably large four-momentum
transfer $Q^{2}$, which corresponds to the squared mass of the $W$ or the $Z$
boson.

The measurements of the inclusive $W$ and $Z$ cross-sections222Throughout this
paper $Z$ includes both the $Z$ and the virtual photon ($\gamma^{\star}$)
contribution. in $pp$ collisions at a centre-of-mass energy of
$7$$\mathrm{\,Te\kern-1.00006ptV}$, using final states containing muons, are
presented in this paper. The analysis is based on data taken by the LHCb
experiment in $2010$ with an integrated luminosity of $37$$\mbox{\,pb}^{-1}$.
The cross-sections are measured in a fiducial region corresponding to the
kinematic coverage of the LHCb detector, where the final state muons have a
transverse momentum, $p^{\mu}_{\mathrm{T}}$, exceeding
$20$${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ and lie within the pseudorapidity
range $2.0<\eta^{\mu}<4.5$. This range is smaller than the LHCb acceptance in
order to avoid edge effects for the acceptance. In addition, the invariant
mass of the muons from the $Z$ boson must be in the range
$60<M_{\mu\mu}<120$${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$. Results are
presented for the total cross-sections and cross-section ratios. Cross-
sections are also measured in bins of muon pseudorapidity for $W$, and in bins
of $Z$ rapidity ($y^{Z}$) for $Z$ production. Because of the presence of the
neutrino, the production asymmetry between $W^{+}$ and $W^{-}$ cannot be
reconstructed as a function of the boson rapidity. Instead it is measured as a
function of the experimentally accessible muon pseudorapidity, $\eta^{\mu}$,
and referred to as the lepton charge asymmetry
$A_{\mu}=(\sigma_{W^{+}\rightarrow\mu^{+}\nu}-\sigma_{W^{-}\rightarrow\mu^{-}\bar{\nu}})/(\sigma_{W^{+}\rightarrow\mu^{+}\nu}+\sigma_{W^{-}\rightarrow\mu^{-}\bar{\nu}})$.
To constrain the PDFs, it is useful to measure $A_{\textmu}$ for different
$p^{\mu}_{\mathrm{T}}$ thresholds. The data are compared to NNLO and NLO pQCD
predictions with recent parametrisations for the PDFs. The signal efficiency
and background contribution are mostly derived from data.

The remainder of the paper is organised as follows. Section 2 describes the
LHCb detector and the Monte Carlo samples. Section 3 describes the selection
of the $W$ and $Z$ candidates, the backgrounds, the determination of the
purity and the signal efficiencies. The measurement of the cross-sections as
well as the systematic uncertainties are discussed in Sect. 4. The results are
presented in Sect. 5 and conclusions in Sect. 6.

##  2 LHCb detector and Monte Carlo samples

The LHCb detector [13] is a single-arm forward spectrometer covering the
pseudorapidity range $2<\eta<5$, designed for the study of particles
containing $b$ or $c$ quarks. The detector includes a high precision tracking
system consisting of a silicon-strip vertex detector (VELO) surrounding the
$pp$ interaction region, a large-area silicon-strip detector (TT) located
upstream of a dipole magnet with a bending power of about $4{\rm\,Tm}$, and
three stations of silicon-strip detectors (IT) and straw drift-tubes (OT)
placed downstream. The combined tracking system has a momentum resolution
$\Delta p/p$ that varies from 0.4% at 5${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$
to 0.6% at 100${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$, and an impact parameter
(IP) resolution of 20$\,\upmu\rm m$ for tracks with high transverse momentum.
Charged hadrons are identified using two ring-imaging Cherenkov detectors.
Photon, electron and hadron candidates are identified by a calorimeter system
consisting of scintillating-pad and pre-shower detectors, an electromagnetic
calorimeter and a hadronic calorimeter. Muons are identified by a muon system
composed of alternating layers of iron and multiwire proportional chambers.
The trigger consists of a hardware stage, based on information from the
calorimeter and muon systems, followed by a software stage which applies a
full event reconstruction. To avoid the possibility that a few events with
high occupancy dominate the CPU time of the software trigger, a set of global
event cuts is applied on the hit multiplicities of most subdetectors used in
the pattern recognition algorithms.

Several Monte Carlo (MC) simulated samples are used to develop the event
selection, estimate the backgrounds, cross-check the efficiencies and to
account for the effect of the underlying event. The Pythia $6.4$ [14]
generator, configured as described in Ref. [15], with the CTEQ6ll [16]
parametrisation for the PDFs is used to simulate the processes
$Z\rightarrow\mu\mu$, $Z\rightarrow\tau\tau$, $W\rightarrow\mu\nu$ and
$W\rightarrow\tau\nu$. The hard partonic interaction is calculated in leading
order pQCD and higher order QCD radiation is modelled using initial and final
state parton showers in the leading log approximation [17]. The fragmentation
into hadrons is simulated in Pythia by the Lund string model [18]. All
generated events are passed through a Geant4 [19] based detector simulation,
the trigger emulation and the event reconstruction chain of the LHCb
experiment.

Samples of $W\rightarrow\mu\nu$ and $Z\rightarrow\mu\mu$ simulated events with
one muon in the LHCb acceptance have been reweighted to reproduce the NNLO
$p^{\mu}_{\mathrm{T}}$ distribution. These samples are referred to as $W$-MC
and $Z$-MC, respectively. In the first step a correction factor is calculated
as a function of the generated muon transverse momentum by determining the
ratio of the generated $p^{\mu}_{\mathrm{T}}$ spectrum, as simulated by the
Powheg [20, *powheg1, *powheg2] generator at NLO, to the generated
$p^{\mu}_{\mathrm{T}}$ spectrum from Pythia. In the second step the events are
reweighted with a factor given by the ratio between the NNLO and NLO
prediction as calculated with Dynnlo [23]. This factor is calculated as a
function of the rapidity of the boson. As an alternative, Pythia samples have
been reweighted to reproduce the $p^{\mu}_{\mathrm{T}}$ distribution as
calculated with Resbos [24, *resbos1, *resbos2]. Resbos includes a NLO
calculation plus next-to-next-to-leading-log resummation of QCD effects at low
transverse momentum.

##  3 Selection of $W$ and $Z$ events

###  3.1 Muon reconstruction and identification

Events with high transverse momentum muons are selected using a single muon
trigger with a threshold of
$p^{\mu}_{\mathrm{T}}>10$${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$. Tracks are
reconstructed starting from the VELO, within which particle trajectories are
approximately straight, since the detector is located upstream of the magnet.
Candidate tracks are extrapolated to the other side of the magnet and a search
is made for compatible hits in the IT and OT sub-detectors. An alternative
strategy searches for track segments in both the VELO and IT/OT detectors and
extrapolates each to the bending plane of the magnet, where they are matched.
Once VELO and IT/OT hits have been combined, an estimate of the track momentum
is available and the full trajectory can be defined. Finally, hits in the TT
sub-detector are added if consistent with the candidate tracks. Thus, the
presence of TT hits can be considered as an independent confirmation of the
validity of the track.

Muons are identified by extrapolating the tracks and searching for compatible
hits in the four most downstream muon stations. For the high momentum muons
that concern this analysis, hits must be found in all four muon stations. In
total, the muon candidate must have passed through over $20$ hadronic
interaction lengths of material.

###  3.2 Selection of $Z\rightarrow\mu\mu$ candidates

Candidate $Z\rightarrow\mu\mu$ events are selected by requiring a pair of well
reconstructed tracks identified as muons; the invariant mass of the two muons
must be in the range
$60<M_{\mu\mu}<120$${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$. Each muon track
must have $p^{\mu}_{\mathrm{T}}>20$${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ and
lie in the range $2.0<\eta^{\mu}<4.5$. The relative uncertainty on the
momentum measurement is required to be less than $10$% and the probability for
the $\chi^{2}/$ndf for the track fit larger than $0.1$%, where ndf is the
number of degrees of freedom. In total, $1966$ $Z$ candidates are selected;
their mass distribution is shown in Fig. 1. The data are not corrected for
initial or final state radiation. A Crystal Ball [27] function for the $Z$
peak, and an exponential distribution for both the off-resonance Drell-Yan
($\gamma^{\star}$) production and the small background contribution are fitted
to the distribution. The fitted mass $90.7\pm
0.1$${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$ and width $3.0\pm
0.1$${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$, where the uncertainties are
statistical, are consistent with expectation from simulation.

Figure 1: Invariant mass of the selected muon pairs. The fitted distribution
to the data is shown as a solid line and the contribution from background and
off-resonance Drell-Yan production as a dashed line.

###  3.3 $Z\rightarrow\mu\mu$ event yield

The background contribution to the $Z\rightarrow\mu\mu$ analysis is very low.
Five different sources are investigated.

  1. 1.

Decays from $Z\rightarrow\tau\tau$ contribute, if both taus decay leptonically
to muons and neutrinos. The tau background is estimated from simulation, with
the $Z$ cross-section fixed to the cross-section measured in this analysis, to
contribute $0.6\pm 0.1$ events to the total sample.

  2. 2.

Decays of heavy flavour hadrons contribute to the background if they decay
semileptonically (“heavy flavour” background). The contribution is estimated
from a sample, which is enriched in background. “Non-isolated” muons are
selected with $p^{\mu}_{\mathrm{T}}>15$${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$
and $M_{\mu\mu}>40$${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$ and the scalar
sum of the transverse momenta of all tracks in a cone of half angle $0.5$ in
$\eta\mbox{-}\phi$ around the muons larger than
$4$${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$; here $\phi$ is the azimuthal angle
measured in radians. A fit to the invariant mass distribution at low masses is
then used to estimate the background contribution in the $Z$ mass region. The
heavy flavour contribution is estimated to be $3.5\pm 0.8$ events.

  3. 3.

Pions or kaons may be misidentified as muons if they decay in flight (“decay-
in-flight” background) or if they travel through the calorimeters and are
identified by the muon chambers (“punch-through” background). This background
should contribute equally in same-sign and opposite-sign combinations of the
muon pair. No event is found in the $Z$ selection with both tracks having the
same charge. The contribution from muon misidentification is estimated to be
less than one event.

  4. 4.

$W$ pair production contributes to the sample if both $W$ bosons decay to a
muon and a neutrino. This contribution corresponds to $0.2\pm 0.1$ events as
estimated with Pythia MC simulation.

  5. 5.

Decays of top quark pairs may contribute if both top quarks decay
semileptonically. Pythia MC simulation predicts a contribution of $0.5\pm 0.2$
events.

The total background contribution in the $Z$ sample in the range
$60\mbox{--}120$${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$ amounts to $4.8\pm
1.0$ events. This corresponds to a purity $\rho^{Z}=0.997\pm 0.001$. The
purity is defined as the ratio of signal to candidate events. No significant
dependence on the boson rapidity is observed.

###  3.4 Selection of $W\rightarrow\mu\nu$ candidates

In leading order QCD, $W\rightarrow\mu\nu$ events are characterised by a
single high transverse momentum muon that is not associated with other
activity in the event. As only the muon can be reconstructed in LHCb, the
background contribution is larger for the $W$ than for the $Z$ candidates.
Therefore, more stringent requirements are placed on the track quality of the
muon and additional criteria are imposed in order to select $W$ candidates.

The optimisation of the $W$ selection and the evaluation of the selection
efficiency make use of a “pseudo-$W$” control sample obtained from the
previously described $Z$ selection, where each of the muons is masked in turn,
in order to mimic the presence of a neutrino and fake a $W\rightarrow\mu\nu$
decay. Excellent agreement is observed for all variables of interest between
pseudo-$W$ and $W$ simulated samples with the exception of those that have an
explicit dependence on the transverse momentum of the muon, as the underlying
momentum distribution differs for muons from $Z$ and $W$.

The identification of $W\rightarrow\mu\nu$ candidate events starts by
requiring a well reconstructed track which is identified as a muon. The track
must have a transverse momentum in the range
$20<p^{\mu}_{\mathrm{T}}<70$${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ within a
pseudorapidity range $2.0<\eta^{\mu}<4.5$. The relative error on the momentum
measurement must be less than $10$%, the probability for the $\chi^{2}/$ndf of
the track fit must be greater than $1$%, and there must be TT hits associated
to the track. The last requirement reduces the number of combinations of VELO
and IT/OT information that have been incorrectly combined to form tracks.

Figure 2:  Distributions for $p_{\mathrm{T}}^{\mathrm{cone}}$ (top) and
$E_{\mathrm{T}}^{\mathrm{cone}}$ (bottom). The points are for muons from
pseudo-$W$ data, the yellow (shaded) histograms are for $W$-MC simulation,
while the open histograms are for muons from QCD background with IP
$>80$$\,\upmu\rm m$ from data. All distributions are normalised to unity.

To suppress background from $Z\rightarrow\mu\mu$ decays, it is required that
any other identified muon in the event has a transverse momentum below
2${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$. This removes the events where both
muons have entered the LHCb acceptance.

Identified muons can originate from background processes of heavy flavour
decays, or misidentification of pions and kaons due to decay-in-flight or
punch-through (“QCD background”). In all such cases, the identified muon is
usually produced in the same direction as the other fragmentation products, in
contrast to muons from $W$ decays which tend to be isolated. The isolation of
the muon is described using the charged transverse momentum,
$p_{\mathrm{T}}^{\mathrm{cone}}$, and neutral transverse energy,
$E_{\mathrm{T}}^{\mathrm{cone}}$, in a cone around the candidate muon. The
quantity $p_{\mathrm{T}}^{\mathrm{cone}}$ is defined as the scalar sum of the
transverse momentum of all tracks, excluding the candidate muon, satisfying
$\sqrt{(\Delta\phi)^{2}+(\Delta\eta^{\mu})^{2}}<0.5$, where $\Delta\phi$ and
$\Delta\eta^{\mu}$ are the differences in $\phi$ and $\eta$ between the muon
candidate and the track. The quantity $E_{\mathrm{T}}^{\mathrm{cone}}$ is
defined in a similar way, but summing the transverse energy of all
electromagnetic calorimeter deposits not associated with tracks. The
distributions for $p_{\mathrm{T}}^{\mathrm{cone}}$ and
$E_{\mathrm{T}}^{\mathrm{cone}}$ are shown in Fig. 2 for pseudo-W data, $W$-MC
and muons with $p_{\mathrm{T}}^{\mu}>20$${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$
and an IP larger than $80$$\,\upmu\rm m$. The IP of the muon is defined as the
distance of closest approach to the primary vertex calculated from the other
tracks in the event excluding the muon candidate. The sample with high IP is
enriched with muons from decays of heavy flavour hadrons, showing the typical
shape of QCD background. There is agreement between pseudo-W data and $W$-MC,
while the shape for the heavy flavour events is quite different. To suppress
QCD background, it is required that
$p_{\mathrm{T}}^{\mathrm{cone}}<2$${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ and
$E_{\mathrm{T}}^{\mathrm{cone}}<2$$\mathrm{\,Ge\kern-1.00006ptV}$.

Muons originating from semi-leptonic decays of heavy flavour hadrons can be
further suppressed by a cut on the IP. Due to the lifetimes of the $B$ and $D$
mesons, these muons do not originate from the primary $pp$ interaction. The IP
distribution is shown in Fig. 4 for pseudo-W events, $W$-MC, and simulated
semi-leptonic decays of hadrons containing a $b$ or $c$ quark. The pseudo-W
events and $W$-MC are in agreement and peak at low values of IP, in contrast
to the heavy flavour background. For the $W$ candidate selection it is
required that IP $<40$$\,\upmu\rm m$. This cut also removes a large fraction
of the background from $W\rightarrow\tau\nu$ and $Z\rightarrow\tau\tau$
decays.

Figure 3:  Muon IP distribution for pseudo-W events as points, $W$-MC as a
yellow (shaded) histogram, and muons from simulated semi-leptonic decays of
hadrons containing a $b$ quark in the full open histogram or a $c$ quark in
the dashed open histogram. All distributions are normalised to unity.

Figure 4: $E/pc$ for pseudo-W events as points, $W$-MC as a yellow (shaded)
histogram, and for hadrons from randomly triggered events in the open
histogram. The energy $E$ is the sum of the energies in the electromagnetic
and hadronic calorimeter associated with the particle. All distributions are
normalised to unity.

Pions and kaons that punch-through to the muon chambers can be distinguished
from true muons as they leave substantial energy deposits in the calorimeters.
Figure 4 shows the summed energy, $E$, in the electromagnetic and hadronic
calorimeter associated with the particle, divided by the track momentum, $p$,
for pseudo-$W$ events, $W$-MC, and hadrons with
$p_{\mathrm{T}}>20$${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ in randomly
triggered events. By requiring $E/pc<0.04$ the punch-through contamination can
be reduced to a negligible level. The disagreement between pseudo-$W$ data and
simulated $W$-MC in Fig. 4 is caused by the different underlying momentum
distribution for muons from $W$ and $Z$.

###  3.5 $W\rightarrow\mu\nu$ event yield

After the $W$ selection requirements are imposed $14\,660$ $W^{+}$ and
$11\,618$ $W^{-}$ candidate events are observed. The $W\rightarrow\mu\nu$
signal yield has been determined by fitting the $p^{\mu}_{\mathrm{T}}$ spectra
of positive and negative muons in data, to template shapes for signal and
backgrounds in five bins of $\eta^{\mu}$. The fit is performed with the
following sources for signal and background with the shapes and normalisations
as described below.

  1. 1.

The $W\rightarrow\mu\nu$ signal template is obtained using the $W$-MC. The
normalisation is left free to vary in each bin of $\eta^{\mu}$ and for each
charge.

  2. 2.

The shape of the template of the largest background, $Z\rightarrow\mu\mu$, is
taken from the $Z$-MC. The normalisation is fixed from data by counting the
number of $Z$ events, scaled by the ratio of events with one muon in the LHCb
acceptance to events with both muons in the acceptance, as determined from
$Z$-MC. The ratio is corrected for the different reconstruction and selection
efficiencies for $W$ and $Z$ as derived from data. This gives an expectation
of $2435\pm 101$ background events (($9.3\pm 0.4$)% of the total sample) in
good agreement with $2335\pm 25$ events found from simulation.

  3. 3.

The shape of the $W\rightarrow\tau\nu$ and $Z\rightarrow\tau\tau$ templates
are taken from Pythia. The $Z\rightarrow\tau\tau$ template is scaled according
to the observed number of $Z$ events. These $\tau$ backgrounds constitute
$2.7$% of the total sample.

  4. 4.

The heavy flavour template is obtained from data by requiring that the muon is
not consistent with originating from the primary vertex (IP $>80$$\,\upmu\rm
m$). The normalisation is determined from data applying all requirements
except for the impact parameter and fitting the resulting IP distribution to
the two templates shown in Fig. 4: the pseudo-$W$ data to describe the signal,
and the simulated heavy flavour events to describe the background. The heavy
flavour contribution is estimated to be ($0.4\pm 0.2$)% of the total sample.

  5. 5.

The punch-through contribution from kaons and pions is largely suppressed by
the requirement on $E/pc$. The $E/pc$ distribution in Fig. 4 is fitted to
pseudo-$W$ data for the signal, and a Gaussian for the punch-through, in order
to estimate the punch-through contribution. This is found to be negligible
($0.02\pm 0.01$)% of the total sample, and also has a shape very similar to
the decay-in-flight component. Hence, this component is not considered when
determining the signal yield.

  6. 6.

The decay-in-flight shape is found from data in a two-step procedure using all
events selected throughout $2010$ by any trigger requirement. First, tracks
with a transverse momentum between $20$ and
$70$${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ are taken to describe the
$p_{\mathrm{T}}$ spectrum of hadrons; tracks that fired a muon trigger are
excluded from the sample. Second, this spectrum is weighted according to the
probability for a hadron to decay-in-flight. This probability is defined as
the fraction of tracks identified as muons in randomly triggered events and is
parametrised as a function of the momentum, $p$, by a function of the form

$1-e^{-\alpha/p},$ (1)

as would be expected for a particle whose mean lifetime in the laboratory
frame scales with its boost. Consistent values for $\alpha$ are found in each
pseudorapidity bin and are in agreement with a calculation of the decay
probability based on the mean lifetimes for charged pions and kaons, and the
distance to the electromagnetic calorimeter before which the hadron must have
decayed. The average of the determinations in each pseudorapidity bin defines
the central value for $\alpha$. The relative normalisation of positively to
negatively charged tracks in each bin of pseudorapidity is fixed to that
observed in randomly triggered events, but the overall normalisation in each
bin of pseudorapidity is left free.

Figure 5:  Distribution of muon $p_{\mathrm{T}}$ for positively (left panel)
and negatively (right panel) charged muons in $W$ candidate events, for the
total fiducial cross-section (a). The plots (b) to (f) give the same
information for the different $\eta^{\mu}$ bins. The data (points) are
compared to the fitted contributions from $W^{-}$ and $W^{+}$ (light shaded).
The background contributions are, from top to bottom in the legend: decay-in-
flight, $Z\rightarrow\mu\mu$, $\tau$ decays of W and $Z$, and heavy flavour
decays.

The default fit has $15$ free parameters: five parameters for the
normalisation of $W^{+}$ in each of the pseudorapidity bins, five parameters
for $W^{-}$, and five parameters for the contribution coming from the decay-
in-flight. The normalisation of the other sources is fixed. The result of the
fit is shown in Fig. 5. Integrated over both charges and
$p^{\mu}_{\mathrm{T}}$ it is found that ($44.3\pm 1.2$)% of the total sample
is due to $W^{+}$, ($34.9\pm 1.1$)% due to $W^{-}$, ($8.5\pm 0.8$)% due to the
decay-in-flight contribution and the remainder due to the other backgrounds.
The $\chi^{2}/$ndf of the fit is $1.002$. The fit is repeated with the
$Z\rightarrow\mu\mu$ and $W\rightarrow\mu\nu$ template corrected with Resbos
instead of NNLO, yielding ($43.6\pm 1.2$)% for $W^{+}$ and ($34.4\pm 1.1$)%
for $W^{-}$ with $\chi^{2}/\mathrm{ndf}=0.983$. The average of the two fits,
which gives a purity $\rho^{W^{+}}=0.788\pm 0.021$ for $W^{+}$ and
$\rho^{W^{-}}=0.784\pm 0.025$ for $W^{-}$, is taken for the final result; half
of the difference is taken as the systematic uncertainty.

##  4 Cross-section measurement

###  4.1 Cross-section definition

Cross-sections are quoted in the kinematical range defined by the
measurements. The cross-sections are measured in bins of $\eta^{\mu}$ for the
$W$ and in bins of $y^{Z}$ in case of the $Z$. The cross-section in a given
bin of $y^{Z}$ ($\eta^{\mu}$) is defined as

$\sigma_{Z\rightarrow\mu\mu}(y^{Z})=\frac{\rho^{Z}f^{Z}_{\mathrm{FSR}}}{\mathcal{L}\mathcal{A}^{Z}}\sum\limits_{\eta_{i}^{\mu},\eta_{j}^{\mu}}{\frac{N^{Z}(\eta_{i}^{\mu},\eta_{j}^{\mu})}{\varepsilon^{Z}(\eta_{i}^{\mu},\eta_{j}^{\mu})}},\mbox{~{}~{}~{}~{}~{}}\sigma_{W\rightarrow\mu\nu}(\eta^{\mu})=\frac{\rho^{W}f^{W}_{\mathrm{FSR}}N^{W}}{\mathcal{L}\mathcal{A}^{W}\varepsilon^{W}},$
(2)

where $N^{Z}(\eta_{i}^{\mu},\eta_{j}^{\mu})$ is the number of $Z$ candidates
in the respective $y^{Z}$ bin with the two muons in the bins $\eta_{i}^{\mu}$
and $\eta_{j}^{\mu}$ being reconstructed with the efficiency
$\varepsilon^{Z}(\eta_{i}^{\mu},\eta_{j}^{\mu})$. Similarly, $N^{W}$ is the
number of $W$ candidates with the muon in the $\eta^{\mu}$ bin. The purity of
the sample ($\rho^{Z(W)}$), the acceptance ($\mathcal{A}^{Z(W)}$), the
correction factor for final state radiation (FSR) ($f^{Z(W)}_{\mathrm{FSR}}$)
and the efficiency ($\varepsilon^{W}$) are determined per bin; $\mathcal{L}$
is the integrated luminosity. The total cross-section is obtained by summing
the contributions of the five $y^{Z}$ or $\eta^{\mu}$ bins.

###  4.2 Signal efficiencies

The data are corrected for efficiency losses due to track reconstruction, muon
identification, and trigger requirements for both analyses. There is an
additional selection efficiency in the $W$ analysis due to the requirements on
the number of additional muons, IP, $E/pc$, $p_{\mathrm{T}}^{\mathrm{cone}}$,
$E_{\mathrm{T}}^{\mathrm{cone}}$, and on TT hits. All efficiencies are
determined from data.

The efficiencies for track reconstruction and muon identification are obtained
using a tag-and-probe method in the $Z$ sample. One of the muons in the $Z$
sample (tag) satisfies all the track criteria. The other muon (probe) is
selected with looser criteria that depend on the efficiency to be measured.
The invariant mass of the dimuon candidates, reconstructed from the tag and
the probe muons, must lie in the window of
$20$${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$ around the nominal $Z$ mass.
The tracking efficiency, which accounts for track reconstruction and the track
quality requirements, is studied using well reconstructed tracks in the muon
stations which are linked to hits in TT. The average track finding efficiency
is about $90$% in the $Z$ sample and about $86$% for the muon in $W$ events.
The tracking efficiency for $W$ is lower due to the more restrictive cuts on
the track quality. The muon identification efficiency is determined with
tracks without the muon identification requirement for the probe muon. The
average single muon efficiency is above $99$%. Both the tracking and the muon
identification efficiencies agree with simulation within errors.

The trigger efficiency contains two components, the first due to the
efficiency of the single muon trigger and the other due to the global event
cuts (GEC). The single muon trigger efficiency is determined using the
$Z\rightarrow\mu\mu$ sample. One muon is required to fire the single muon
trigger. The trigger response of the other muon then defines the trigger
efficiency. The requirement on the occupancy of the events depends on the
multiplicity of the primary interactions. It was checked with a sample which
did not have the GEC applied, that no events are lost if there is only one
primary vertex reconstructed. The GEC efficiency as a function of the number
of primary vertices is determined by adding randomly triggered events to
events with $Z$ ($W$) candidates with one primary vertex; on average it
amounts to $93$%. The overall trigger efficiency is calculated for each event
depending on the lepton pseudorapidity and the primary vertex multiplicity. It
is found to be about $88$% for the $Z$ and $75$% for the $W$ sample.

The $W$ selection requires that there are no other muons with
$p^{\mu}_{\mathrm{T}}>2$${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$,
$p_{\mathrm{T}}^{\mathrm{cone}}<2$${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$,
$E_{\mathrm{T}}^{\mathrm{cone}}<2$$\mathrm{\,Ge\kern-1.00006ptV}$, IP
$<40$$\,\upmu\rm m$, and $E/pc<0.04$. The selection efficiency is determined
from the fraction of pseudo-W events that pass these requirements. A similar
method is used to evaluate the efficiency for the requirement of TT hits
associated to the muon track of the $W$ candidate. Simulation studies show
that with the exception of the $E/pc$ distribution, the pseudo-W data provide
a consistent description of $W\rightarrow\mu\nu$ simulation, as shown in Figs.
2 and 4. However, the harder muon $p_{\mathrm{T}}$ spectrum in pseudo-W data
leads to slightly lower values of $E/pc$ than for muons produced in $W$
decays. The simulation is used to determine this difference, which is only
significant for $\eta^{\mu}$ between $2.0$ and $2.5$, where the efficiency for
$W$ events is estimated to be $2.1$% lower than for pseudo-W data. The
selection efficiency is about $67$% for $2.5<\eta^{\mu}<4.0$ and drops to
about $53$% and $33$% for the two bins at the edge of the acceptance with
$2.0<\eta^{\mu}<2.5$ and $4.0<\eta^{\mu}<4.5$, respectively.

All the efficiencies have been checked for possible dependences on
$p_{\mathrm{T}}^{\mu}$, the azimuthal angle of the muon, magnet polarity, and
$\eta^{\mu}$. Only the latter exhibits a significant dependence, which is
taken into account. Since any charge bias of the efficiencies would directly
influence the measurement of the lepton charge asymmetry, it was checked there
is no significant charge dependence within the uncertainties of the
efficiencies. The efficiency corrections are applied as a function of the
pseudorapidity of the muons except the GEC.

The efficiencies are uncorrelated between pseudorapidity bins but correlated
for $W^{+}$, $W^{-}$ and $Z$. These correlations are taken into account for
the measurement of the lepton charge asymmetry and the cross-section ratios.

###  4.3 Acceptance

The selection criteria for the $W$ and $Z$ define the fiducial region of the
measurement. Simulated events are used to determine the acceptance
$\mathcal{A}$, defined as $\mathcal{A}={N_{\mathrm{rec}}}/{N_{\mathrm{gen}}}$.
Here, $N_{\mathrm{rec}}$ is the number of reconstructed events satisfying the
cuts on the pseudorapidity and the minimal momenta of the reconstructed muons,
as well as on the dimuon mass in the case of the $Z$ analysis. Similarly,
$N_{\mathrm{gen}}$ is the number of generated events with the cuts applied on
the generated muons. The acceptance is estimated with $W$-MC and $Z$-MC. It is
found to be consistent with unity for the $Z$ and above $0.99$ for the $W$
analysis. For the latter, the acceptance corrects for the small loss of events
with $p^{\mu}_{\mathrm{T}}>70$${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$.

###  4.4 Luminosity

The absolute luminosity scale was measured at specific periods during the data
taking, using both Van der Meer scans [28] where colliding beams are moved
transversely across each other to determine the beam profile, and a beam-gas
imaging method [29, 30]. For the latter, reconstructed beam-gas interaction
vertices near the beam crossing point determine the beam profile. Both methods
give similar results and are estimated to have a precision of order $3.5$%.
The knowledge of the absolute luminosity scale is used to calibrate the number
of tracks in the VELO, which is found to be stable throughout the data-taking
period and can therefore be used to monitor the instantaneous luminosity of
the entire data sample. The dataset for this analysis corresponds to an
integrated luminosity of $37.1\pm 1.3$$\mbox{\,pb}^{-1}$.

###  4.5 Corrections to the data

The measured cross-sections are corrected to Born level in quantum
electrodynamics (QED) in order to provide a consistent comparison with NLO and
NNLO QCD predictions, which do not include the effects of FSR. Corrections
have been estimated using Photos [31] interfaced to Pythia. The Pythia
$p_{\mathrm{T}}$ spectrum of the electroweak boson has been reweighted to the
NNLO spectrum as determined with Dynnlo [23]. The correction is taken as the
number of events within the fiducial cuts of the measurements after FSR
divided by the number of events generated within the fiducial cuts.

Pythia simulation is used to study bin-to-bin migrations for $\eta^{\mu}$ and
$y^{Z}$. No significant net migration is observed and no correction is
applied.

###  4.6 Systematic uncertainties

Table 1: Contributions to the systematic uncertainty for the total $Z$ and $W$
cross-sections. The different contributions are discussed in Sect. 4.6

Source |  $\Delta\sigma_{Z\rightarrow\mu\mu}$ (%) |  $\Delta\sigma_{W^{+}\rightarrow\mu^{+}\nu}$ (%) |  $\Delta\sigma_{W^{-}\rightarrow\mu^{-}\bar{\nu}}$ (%)
---|---|---|---
Signal purity | $\pm 0.1$ | $\pm 1.2$ | $\pm 0.9$
Template shape (fit) | – | $\pm 0.9$ | $\pm 1.0$
Efficiency (trigger, tracking, muon id) | $\pm 4.3$ | $\pm 2.2$ | $\pm 2.0$
Additional selection | – | $\pm 1.8$ | $\pm 1.7$
FSR correction | $\pm 0.02$ | $\pm 0.01$ | $\pm 0.02$
Total | $\pm 4.3$ | $\pm 3.2$ | $\pm 2.9$
Luminosity | $\pm 3.5$ | $\pm 3.5$ | $\pm 3.5$

Aside from the uncertainty on the luminosity measurement, the main sources of
experimental uncertainties come from the efficiency determinations and the
background estimate in the $W$ analysis. The following sources have been
considered:

  1. 1.

The relative uncertainties of the tracking, muon identification, trigger and
GEC efficiencies are added in quadrature. They lead to a systematic
uncertainty for the total cross-sections of $4.3$% ($2.2$%, $2.0$%) for the
$Z$ ($W^{+}$, $W^{-}$).

  2. 2.

The statistical uncertainty on the efficiency of the additional selection cuts
for the $W$ analysis translates into a $1.8$% ($1.7$%) systematic uncertainty
on the total $W^{+}$ ($W^{-}$) cross-section.

  3. 3.

The uncertainty of the background contribution for the $Z$ analysis is small;
the uncertainty in the determination of the sample purity leads to a $0.1$%
uncertainty on the total cross-section.

  4. 4.

Both the shape and normalisations of the templates used in the $W$ fit are
considered as an additional source of uncertainty. To determine this
systematic uncertainty each of the following sources is varied in turn, the
data are refitted to determine the fraction of $W^{+}$ and $W^{-}$ events, and
the deviations from the original signal yield are combined in quadrature. The
following variations are made:

     * •

the difference of the two fits using different $W$ and $Z$ templates (see
Sect. 3.5) leads to a variation on the $W$ fractions of $0.8$%;

     * •

the normalisation of the $Z$ component was changed by the statistical
uncertainty with which it was determined, leading to a variation in the $W$
fractions of $0.3$%;

     * •

the normalisation of the $W\rightarrow\tau\nu$ template was changed by the
statistical uncertainty with which it was determined, leading to a negligible
change in the $W$ fractions, since this template shape is very similar to the
decay-in-flight template which is allowed to vary in the fit;

     * •

the heavy flavour template has also been changed by the statistical
uncertainty with which it was determined leading to a negligible change in the
$W$ fractions;

     * •

instead of leaving the relative normalisation of the decays-in-flight template
between pseudorapidity bins to be free in the fit, this is fixed to the values
observed in randomly triggered events, and the full fit performed with a
single free parameter for the background; the $W$ fractions change by $0.2$%;

     * •

the shape of the decay-in-flight template has been changed using different
values for $\alpha$ (see Eq. 1) to describe the decay probability,
corresponding to different regions in which the hadron must have
decayed;333Three different decay regions have been considered: from the
interaction point, from the VELO and from the TT stations up to the
electromagnetic calorimeter. no difference in the $W$ fractions is observed.

  5. 5.

The uncertainty on the FSR correction is evaluated for each bin as the maximum
of the statistical uncertainty of the correction factor and the difference
between the weighted and unweighted FSR correction factor.

The sources of systematic uncertainties are summarised in Table 1, together
with the size of the resultant uncertainty on the $W$ and $Z$ total cross-
sections. The total systematic uncertainty is the sum of all contributions
added in quadrature.

##  5 Results

The inclusive cross-sections for $Z\rightarrow\mu\mu$ and $W\rightarrow\mu\nu$
production for muons with
$p^{\mu}_{\mathrm{T}}>20$${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ in the
pseudorapidity region $2.0<\eta^{\mu}<4.5$ and, in the case of $Z$, the
invariant mass range
$60<M_{\mu\mu}<120$${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$ are measured to
be

$\sigma_{Z\rightarrow\mu\mu}$ | = |  $76.7\pm 1.7\pm 3.3\pm 2.7$$\rm\,pb$
---|---|---
$\sigma_{W^{+}\rightarrow\mu^{+}\nu}$ | = | $831\pm 9\pm 27\pm 29$$\rm\,pb$
$\sigma_{W^{-}\rightarrow\mu^{-}\bar{\nu}}$ | = |  $656\pm 8\pm 19\pm 23$$\rm\,pb$ ,

Figure 6: Measurements of the $Z$, $W^{+}$ and $W^{-}$ cross-section and
ratios, data are shown as bands which the statistical (dark shaded/orange) and
total (light hatched/yellow) errors. The measurements are compared to NNLO and
NLO predictions with different PDF sets for the proton, shown as points with
error bars. The PDF uncertainty, evaluated at the $68$% confidence level, and
the theoretical uncertainties are added in quadrature to obtain the
uncertainties of the predictions.

where the first uncertainty is statistical, the second systematic and the
third is due to the luminosity. All the measurements are dominated by the
luminosity and the systematic uncertainty. The latter is dominated by the
limited number of events for the background templates and in the determination
of the efficiencies.

The ratios
$R_{W}=\sigma_{W^{+}\rightarrow\mu^{+}\nu}/\sigma_{W^{-}\rightarrow\mu^{-}\bar{\nu}}$
and
$R_{WZ}=(\sigma_{W^{+}\rightarrow\mu^{+}\nu}+\sigma_{W^{-}\rightarrow\mu^{-}\bar{\nu}})/\sigma_{Z\rightarrow\mu\mu}$
are measured to be

$R_{W}$ | = | $1.27\pm 0.02\pm 0.01$
---|---|---
$R_{WZ}$ | = |  $19.4\pm 0.5\pm 0.9$ .

Here, the uncertainty from the luminosity completely cancels. The systematic
uncertainty on the trigger, muon identification, tracking and selection
efficiencies, as well as the uncertainty on the purity are assumed to be fully
correlated between $W^{+}$ and $W^{-}$. No correlation is assumed between the
$\eta^{\mu}$ bins, except for the purity. The uncertainty on the $Z$ cross-
section from the reconstruction efficiency is correlated between boson
rapidity bins. The correlation of the uncertainty on the efficiencies between
$W$ and $Z$ are estimated with MC simulation to be $90$%. The full correlation
matrix is given in the Appendix (Table 2). The ratio of the $W$ to $Z$ cross-
section is measured, for each charge separately, to be

$\sigma_{W^{+}\rightarrow\mu^{+}\nu}/\sigma_{Z\rightarrow\mu\mu}$ | = | $10.8\pm 0.3\pm 0.5$
---|---|---
$\sigma_{W^{-}\rightarrow\mu^{-}\bar{\nu}}/\sigma_{Z\rightarrow\mu\mu}$ | = |  $8.5\pm 0.2\pm 0.4$ .

A summary of the measurements of the inclusive cross-sections
$\sigma_{W^{+}\rightarrow\mu^{+}\nu}$,
$\sigma_{W^{-}\rightarrow\mu^{-}\bar{\nu}}$ and $\sigma_{Z\rightarrow\mu\mu}$,
and the ratios is shown in Fig. 6. The measurements are shown as a band which
represents the total and statistical uncertainties.

Figure 7: Differential cross-section for $Z\rightarrow\mu\mu$ as a function of
$y^{Z}$. The dark shaded (orange) bands correspond to the statistical
uncertainties, the light hatched (yellow) band to the statistical and
systematic uncertainties added in quadrature. Superimposed are NNLO (NLO)
predictions with different parametrisations for the PDF as points with error
bars; they are displaced horizontally for presentation.

Figure 8: Differential $W$ cross-section in bins of muon pseudorapidity. The
dark shaded (orange) bands correspond to the statistical uncertainties, the
light hatched (yellow) band to the statistical and systematic uncertainties
added in quadrature. Superimposed are NNLO (NLO) predictions as described in
Fig 8.  Figure 9: Lepton charge asymmetry
$A_{\mu}=(\sigma_{W^{+}\rightarrow\mu^{+}\nu}-\sigma_{W^{-}\rightarrow\mu^{-}\bar{\nu}})/(\sigma_{W^{+}\rightarrow\mu^{+}\nu}+\sigma_{W^{-}\rightarrow\mu^{-}\bar{\nu}})$
in bins of muon pseudorapidity. The dark shaded (orange) bands correspond to
the statistical uncertainties, the light hatched (yellow) band to the
statistical and systematic uncertainties added in quadrature. Superimposed are
NNLO (NLO) predictions as described in Fig 8. The MSTW08 values for
$\eta^{\mu}<2$ represent the central value of the prediction.

The results are compared to theoretical predictions calculated at NNLO with
the program Dynnlo [23] for the NNLO PDF sets of MSTW08 [32], ABKM09 [33],
JR09 [34], HERA15 [12] and NNPDF21 [35] and at NLO for the NLO PDF set CTEQ6m
[16].444Dynnlo sets $\alpha_{s}$ to the value of $\alpha_{s}$ at the mass of
the $Z$ boson as given by the different PDF sets. The scale uncertainties are
estimated by varying the renormalisation and factorisation scales by factors
of two around the nominal value, which is set to the boson mass. The
uncertainties for each set correspond to the PDF uncertainties at 68% and the
scale uncertainties added in quadrature.555The uncertainties for the PDF set
from CTEQ6m which is given at $90$% CL are divided by $1.645$.

While the $W^{-}$ and $Z$ cross-sections are well described by all
predictions, the $W^{+}$ cross-section is slightly overestimated by the ABKM09
and NNPDF21 PDF sets. The ratio of the $W^{-}$ to $Z$ cross-sections agrees
reasonably well with the predictions, but the $W^{+}$ to the $Z$ ratio is
overestimated by most of the predictions. The systematic uncertainties for the
$R_{W}$ almost cancel and also the theoretical uncertainties are much reduced.
The $R_{W}$ measurement tests the Standard Model predictions with a precision
of $1.7\%$ which is comparable to the uncertainty of the theoretical
prediction. The ABKM09 prediction overestimates this ratio while all the other
predictions agree with the measurement.

Figure 10:
$R_{W}=\sigma_{W^{+}\rightarrow\mu^{+}\nu}/\sigma_{W^{-}\rightarrow\mu^{-}\bar{\nu}}$
in bins of muon pseudorapidity. The dark shaded (orange) bands correspond to
the statistical uncertainties, the light hatched (yellow) band to the
statistical and systematic uncertainties added in quadrature. Superimposed are
NNLO (NLO) predictions with different parametrisations as described in Fig 8.
The MSTW08 values for $\eta^{\mu}<2$ represent the central value of the
prediction.

Differential distributions are measured in five bins in $y^{Z}$ for the Z and
of $\eta^{\mu}$ for the W. Figure 8 shows the differential cross-section as a
function of the rapidity of the $Z$ boson together with NNLO (NLO) predictions
with different parametrisation for the PDFs of the proton. The predictions
agree with the measurements within uncertainties though all the predictions
are lower than the measured cross-section for $2.5<\eta^{\mu}<3.0$. The
differential cross-sections are listed in Table 4 in the Appendix.

The differential distribution of the $W^{+}$ and $W^{-}$ cross-section, the
lepton charge asymmetry $A_{\mu}$ and the ratio $R_{W}$ as a function of the
muon pseudorapidity are shown in Figs. 8, 9 and 10 and listed in Tables 4 to 6
as a function of $p^{\mu}_{\mathrm{T}}$. The measurement of the charge
asymmetry and the $W$ ratio provides important additional information on the
PDFs particularly on the valence quark distributions [11].

Since the inclusive cross-section for $W^{+}$ is larger than for $W^{-}$, due
to the excess of $u$ over $d$ quarks in the proton, the overall asymmetry is
positive. The asymmetry and the $W$ cross-sections strongly vary as a function
of the pseudorapidity of the charged lepton, and $A_{\mu}$ even changes sign,
owing to differing helicity dependence of the lepton couplings to the boson.
This behaviour is reflected in the differential $W$ cross-sections, where at
large muon pseudorapidities the $W^{-}$ cross-section is higher than the
$W^{+}$ cross-section, as a consequence of the $V-A$ structure of the $W$ to
lepton coupling. The cross-section and the asymmetry measurements are compared
to the NNLO (NLO) predictions with different parameterisation for the PDFs.
The ABKM09 prediction overestimates the measured asymmetry in three of the
five bins. The other predictions describe the measurement within
uncertainties.

Figure 11: Lepton charge asymmetry
$A_{\mu}=(\sigma_{W^{+}\rightarrow\mu^{+}\nu}-\sigma_{W^{-}\rightarrow\mu^{-}\bar{\nu}})/(\sigma_{W^{+}\rightarrow\mu^{+}\nu}+\sigma_{W^{-}\rightarrow\mu^{-}\bar{\nu}})$
for muons with $p^{\mu}_{\mathrm{T}}>$25 (top) and
30${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ (bottom), respectively in bins of
muon pseudorapidity. The dark shaded (orange) bands correspond to the
statistical uncertainties, the light hatched (yellow) band to the statistical
and systematic uncertainties added in quadrature. The statistical uncertainty
is undistinguishable from the total uncertainty. Superimposed are the NNLO
predictions with the MSTW08 parametrisation for the PDF.

The asymmetry is also measured for two higher $p^{\mu}_{\mathrm{T}}$
thresholds for the muons, at 25 and 30${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$.
The result is shown in Fig. 11 and listed in Table 6. The NNLO prediction with
MSTW08 parametrisation for the PDF also describes the measured asymmetry with
the higher cuts on the transverse momentum of the muon.

##  6 Conclusions

Measurements of inclusive $W$ and $Z$ boson production in $pp$ collisions at
$\sqrt{s}=7$$\mathrm{\,Te\kern-1.00006ptV}$ with final states containing muons
have been performed using $37$$\mbox{\,pb}^{-1}$ of data collected with the
LHCb detector. The inclusive cross-sections have been measured separately for
$W^{+}$ and $W^{-}$ production as well as the ratios
$\sigma_{W^{+}\rightarrow\mu^{+}\nu}/\sigma_{W^{-}\rightarrow\mu^{-}\bar{\nu}}$
and
$(\sigma_{W^{+}\rightarrow\mu^{+}\nu}+\sigma_{W^{-}\rightarrow\mu^{-}\bar{\nu}})/\sigma_{Z\rightarrow\mu\mu}$
and the lepton charge asymmetry
$(\sigma_{W^{+}\rightarrow\mu^{+}\nu}-\sigma_{W^{-}\rightarrow\mu^{-}\bar{\nu}})/(\sigma_{W^{+}\rightarrow\mu^{+}\nu}+\sigma_{W^{-}\rightarrow\mu^{-}\bar{\nu}})$.
The results have been compared to five next-to-next-to-leading order QCD
predictions with different sets for the parton density functions of the proton
and to one calculation at next-to-leading order. There is general agreement
with the predictions, though some of the PDF sets overestimate the ratios of
the cross-sections. The ratio
$\sigma_{W^{+}\rightarrow\mu^{+}\nu}/\sigma_{W^{-}\rightarrow\mu^{-}\bar{\nu}}=$$1.27\pm
0.02\pm 0.01$ is measured precisely and allows the Standard Model prediction
to be tested with an accuracy of about $1.7$%, comparable to the uncertainty
on the theory prediction. These represent the first measurements of the $W$
and $Z$ production cross-sections and ratios in the forward region at the LHC,
and will provide valuable input to the knowledge of the parton density
functions of the proton. The uncertainty on the cross-section measurements is
dominated by systematic uncertainties. Since most of these are statistical in
nature, the accuracy on the measurement with further data is expected to
significantly improve.

## Acknowledgements

We express our gratitude to our colleagues in the CERN accelerator departments
for the excellent performance of the LHC. We thank the technical and
administrative staff at CERN and at the LHCb institutes, and acknowledge
support from the National Agencies: CAPES, CNPq, FAPERJ and FINEP (Brazil);
CERN; NSFC (China); CNRS/IN2P3 (France); BMBF, DFG, HGF and MPG (Germany); SFI
(Ireland); INFN (Italy); FOM and NWO (The Netherlands); SCSR (Poland); ANCS
(Romania); MinES of Russia and Rosatom (Russia); MICINN, XuntaGal and GENCAT
(Spain); SNSF and SER (Switzerland); NAS Ukraine (Ukraine); STFC (United
Kingdom); NSF (USA). We also acknowledge the support received from the ERC
under FP7 and the Region Auvergne.

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Appendix

##  Appendix A Tables of results

Table 2: Correlation coefficients between $W^{+}$, $W^{-}$ and Z in the five bins considered. The luminosity uncertainty is not included. | $2<\eta^{\mu}\,(y^{Z})<2.5$ | $2.5<\eta^{\mu}\,(y^{Z})<3$ | $3<\eta^{\mu}\,(y^{Z})<3.5$ | $3.5<\eta^{\mu}\,(y^{Z})<4$ | $4<\eta^{\mu}\,(y^{Z})<4.5$ |
---|---|---|---|---|---|---
$W^{+}$ | 1 |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  $2<\eta^{\mu}\,(y^{Z})<2.5$
$W^{-}$ | 0.87 | 1 |  |  |  |  |  |  |  |  |  |  |  |  |
$Z$ | 0.36 | 0.34 | 1 |  |  |  |  |  |  |  |  |  |  |  |
$W^{+}$ | 0.02 | 0.02 | 0.35 | 1 |  |  |  |  |  |  |  |  |  |  |  |  $2.5<\eta^{\mu}\,(y^{Z})<3$
$W^{-}$ | 0.02 | 0.02 | 0.35 | 0.90 | 1 |  |  |  |  |  |  |  |  |  |
$Z$ | 0.47 | 0.44 | 0.45 | 0.45 | 0.45 | 1 |  |  |  |  |  |  |  |  |
$W^{+}$ | 0.02 | 0.03 | 0.24 | 0.02 | 0.02 | 0.31 | 1 |  |  |  |  |  |  |  |  |  $3<\eta^{\mu}\,(y^{Z})<3.5$
$W^{-}$ | 0.02 | 0.02 | 0.29 | 0.02 | 0.02 | 0.37 | 0.89 | 1 |  |  |  |  |  |  |
$Z$ | 0.46 | 0.43 | 0.44 | 0.45 | 0.44 | 0.58 | 0.31 | 0.37 | 1 |  |  |  |  |  |
$W^{+}$ | 0.04 | 0.05 | 0.35 | 0.04 | 0.04 | 0.45 | 0.05 | 0.04 | 0.44 | 1 |  |  |  |  |  |  $3.5<\eta^{\mu}\,(y^{Z})<4$
$W^{-}$ | 0.02 | 0.02 | 0.40 | 0.02 | 0.01 | 0.52 | 0.02 | 0.02 | 0.51 | 0.80 | 1 |  |  |  |
$Z$ | 0.32 | 0.29 | 0.30 | 0.30 | 0.30 | 0.39 | 0.21 | 0.25 | 0.39 | 0.30 | 0.35 | 1 |  |  |
$W^{+}$ | 0.07 | 0.09 | 0.19 | 0.07 | 0.07 | 0.24 | 0.09 | 0.07 | 0.24 | 0.15 | 0.06 | 0.16 | 1 |  |  |  $4<\eta^{\mu}\,(y^{Z})<4.5$
$W^{-}$ | 0.01 | 0.01 | 0.28 | 0.01 | 0.01 | 0.37 | 0.01 | 0.01 | 0.36 | 0.02 | 0.01 | 0.24 | 0.57 | 1 |
$Z$ | 0.03 | 0.03 | 0.03 | 0.03 | 0.03 | 0.04 | 0.02 | 0.03 | 0.04 | 0.03 | 0.04 | 0.03 | 0.02 | 0.03 | 1
| $W^{+}$ | $W^{-}$ | $Z$ | $W^{+}$ | $W^{-}$ | $Z$ | $W^{+}$ | $W^{-}$ | $Z$ | $W^{+}$ | $W^{-}$ | $Z$ | $W^{+}$ | $W^{-}$ | $Z$ |
Table 3: Differential $Z\rightarrow\mu\mu$ cross-section,
$d\sigma_{Z\rightarrow\mu\mu}/dy^{Z}$, in bins of boson rapidity. The first
cross-section uncertainty is statistical, the second systematic, and the third
due to the uncertainty on the luminosity determination. The correction factor
$f^{Z}_{\mathrm{FSR}}$ which is used to correct for FSR is listed separately.

$y^{Z}$ | $d\sigma_{Z\rightarrow\mu\mu}/dy^{Z}$ [pb] | $f^{Z}_{\mathrm{FSR}}$
---|---|---
$2.0-2.5$ | $25.5$ | $\pm 1.4$ | $\pm 1.0$ | $\pm 0.9$ | $1.020\pm 0.001$
$2.5-3.0$ | $66.8$ | $\pm 2.3$ | $\pm 2.7$ | $\pm 2.3$ | $1.018\pm 0.001$
$3.0-3.5$ | $49.8$ | $\pm 2.0$ | $\pm 2.2$ | $\pm 1.7$ | $1.018\pm 0.001$
$3.5-4.0$ | $11.1$ | $\pm 0.9$ | $\pm 0.6$ | $\pm 0.4$ | $1.024\pm 0.001$
$4.0-4.5$ | $0.074$ | $\pm 0.074$ | $\pm 0.004$ | $\pm 0.002$ | $1.027\pm 0.027$

Table 4: Differential $W\rightarrow\mu\nu$ cross-section,
$d\sigma_{W\rightarrow\mu\nu}/\eta^{\mu}$, in bins of lepton pseudorapidity.
The first cross-section uncertainty is statistical, the second systematic, and
the third due to the uncertainty on the luminosity determination. The
correction factor $f^{W}_{\mathrm{FSR}}$ which is used to correct for FSR is
listed separately.

| $\eta^{\mu}$ |  $d\sigma_{W\rightarrow\mu\nu}/\eta^{\mu}$ [pb] | $f^{W}_{\mathrm{FSR}}$
---|---|---|---
$W^{+}$ | $2.0-2.5$ | $691$ | $\pm 12$ | $\pm 37$ | $\pm 24$ | $1.0146\pm 0.0004$
| $2.5-3.0$ | $530$ | $\pm 9$ | $\pm 30$ | $\pm 19$ | $1.0086\pm 0.0002$
| $3.0-3.5$ | $296$ | $\pm 7$ | $\pm 23$ | $\pm 10$ | $1.0107\pm 0.0006$
| $3.5-4.0$ | $121$ | $\pm 5$ | $\pm 19$ | $\pm 4$ | $1.0097\pm 0.0005$
| $4.0-4.5$ | $23.1$ | $\pm 3.2$ | $\pm 4.9$ | $\pm 0.8$ | $1.0009\pm 0.0009$
$W^{-}$ | $2.0-2.5$ | $393$ | $\pm 9$ | $\pm 22$ | $\pm 13$ | $1.0147\pm 0.0008$
| $2.5-3.0$ | $370$ | $\pm 8$ | $\pm 20$ | $\pm 13$ | $1.0163\pm 0.0004$
| $3.0-3.5$ | $282$ | $\pm 7$ | $\pm 18$ | $\pm 10$ | $1.0147\pm 0.0004$
| $3.5-4.0$ | $200$ | $\pm 6$ | $\pm 14$ | $\pm 7$ | $1.0173\pm 0.0008$
| $4.0-4.5$ | $68$ | $\pm 5$ | $\pm 10$ | $\pm 2$ | $1.0194\pm 0.0009$

Table 5: Lepton charge asymmetry, $A_{\mu}$, in bins of muon pseudorapidity
for a $p^{\mu}_{\mathrm{T}}$ threshold at 20, 25 and 30 GeV/c. The first
uncertainty is statistical and the second systematic. The effect of FSR is at
the level of $10^{-4}$ and is not listed.

$\eta^{\mu}$ |  $A_{\mu}$ ($p_{\mathrm{T}}^{\mu}>$20${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$) |  $A_{\mu}$ ($p_{\mathrm{T}}^{\mu}>$25${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$) |  $A_{\mu}$ ($p_{\mathrm{T}}^{\mu}>$30${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$)
---|---|---|---
$2.0-2.5$ | $0.275$ | $\pm 0.014$ | $\pm 0.003$ | $0.256$ | $\pm 0.015$ | $\pm 0.002$ | $0.238$ | $\pm 0.018$ | $\pm 0.002$
$2.5-3.0$ | $0.178$ | $\pm 0.013$ | $\pm 0.002$ | $0.195$ | $\pm 0.015$ | $\pm 0.001$ | $0.219$ | $\pm 0.017$ | $\pm 0.001$
$3.0-3.5$ | $0.024$ | $\pm 0.016$ | $\pm 0.009$ | $0.054$ | $\pm 0.018$ | $\pm 0.003$ | $0.112$ | $\pm 0.022$ | $\pm 0.002$
$3.5-4.0$ | $-0.247$ | $\pm 0.022$ | $\pm 0.011$ | $-0.203$ | $\pm 0.027$ | $\pm 0.005$ | $-0.124$ | $\pm 0.035$ | $\pm 0.003$
$4.0-4.5$ | $-0.493$ | $\pm 0.058$ | $\pm 0.051$ | $-0.413$ | $\pm 0.081$ | $\pm 0.016$ | $-0.353$ | $\pm 0.122$ | $\pm 0.008$

Table 6: $W$ cross-section ratio,
$R_{W}=\sigma_{W^{+}\rightarrow\mu^{+}\nu}/\sigma_{W^{-}\rightarrow\mu^{-}\bar{\nu}}$,
in bins of muon pseudorapidity. The first error is statistical and the second
systematic. The effect of FSR is at the level of $10^{-4}$ and is not listed.

$\eta^{\mu}$ | $R_{W}$
---|---
$2.0-2.5$ | $1.76$ | $\pm 0.05$ | $\pm 0.01$
$2.5-3.0$ | $1.43$ | $\pm 0.04$ | $\pm 0.01$
$3.0-3.5$ | $1.05$ | $\pm 0.03$ | $\pm 0.02$
$3.5-4.0$ | $0.60$ | $\pm 0.03$ | $\pm 0.01$
$4.0-4.5$ | $0.34$ | $\pm 0.05$ | $\pm 0.05$