arX

iv:0

907.

3713

v2 [

hep-

ex]

11

Oct

200

9

A novelm ethod form odeling therecoilin W boson eventsathadron

colliders

V.M .Abazovak,B.Abbottbw ,M .Abolinsbm ,B.S.Acharyaad,M .Adam say,T.Adam saw ,E.Aguilof,

M .Ahsanbg,G .D.Alexeevak,G .Alkhazovao,A.Altonbl,1,G .Alversonbk,G .A.Alvesb,L.S.Ancuaj,

T.Andeenba,M .S.Anzelcba,M .Aokiax,Y.Arnoudn,M .Arovbh,M .Arthaudr,A.Askewaw ,2,B.�Asm anap,

O .Atram entovaw ,2,C.Avilah,J.BackusM ayescd,F.Badaudm ,L.Bagbyax,B.Baldinax,D.V.Bandurinbg,

S.Banerjeead,E.Barberisbk,A.-F.Barfusso,P.Bargassacb,P.Baringerbf,J.Barretob,J.F.Bartlettax,

U.Basslerr,D.Bauerar,S.Bealef,A.Beanbf,M .Begallic,M .Begelbu,C.Belanger-Cham pagneap,

L.Bellantoniax,A.Bellavanceax,J.A.Benitezbm ,S.B.Beriab,G .Bernardiq,R.Bernhardw ,I.Bertram aq,

M .Besan�conr,R.Beuselinckar,V.A.Bezzubovan,P.C.Bhatax,V.Bhatnagarab,G .Blazeyaz,S.Blessingaw ,

K .Bloom bo,A.Boehnleinax,D.Bolinebj,T.A.Boltonbg,E.E.Boosam ,G .Borissovaq,T.Bosebj,

A.Brandtbz,R.Brockbm ,G .Brooijm ansbr,A.Brossax,D.Browns,X.B.Bug,D.Buchholzba,

M .Buehlercc,V.Buescherv,V.Bunichevam ,S.Burdinaq,3,T.H.Burnettcd,C.P.Buszelloar,P.Calfayanz,

B.Calpaso,S.Calvetp,J.Cam m inbs,M .A.Carrasco-Lizarragaah,E.Carreraaw,W .Carvalhoc,

B.C.K .Caseyax,H.Castilla-Valdezah,S.Chakrabartibt,D.Chakrabortyaz,K .M .Chanbc,A.Chandraav,

E.Cheuat,D.K .Chobj,S.W .Choaf,S.Choiag,B.Choudharyac,T.Christoudiasar,S.Cihangirax,

D.Claesbo,J.Clutterbf,M .Cookeax,W .E.Cooperax,M .Corcorancb,F.Coudercr,M .-C.Cousinouo,

D.Cuttsby,M .�Cwiokae,A.Dasat,G .Daviesar,K .Debz,S.J.de Jongaj,E.De La Cruz-Bureloah,

K .DeVaughanbo,F.D�eliotr,M .Dem arteauax,R.Dem inabs,D.Denisovax,S.P.Denisovan,S.Desaiax,

H.T.Diehlax,M .Diesburgax,A.Dom inguezbo,T.Dorlandcd,A.Dubeyac,L.V.Dudkoam ,L.Du otp,

D.Dugganaw ,A.Duperrino,S.Duttab,A.Dyshkantaz,M .Eadsbo,D.Edm undsbm ,J.Ellisonav,

V.D.Elviraax,Y.Enariby,S.Enobi,M .Escaliero,H.Evansbb,A.Evdokim ovbu,V.N.Evdokim ovan,

G .Facinibk,A.V.Ferapontovbg,T.Ferbelbi,bs,F.Fiedlery,F.Filthautaj,W .Fisherax,H.E.Fiskax,

M .Fortneraz,H.Foxaq,S.Fuax,S.Fuessax,T.G adfortbr,C.F.G aleaaj,A.G arcia-Bellidobs,V.G avriloval,

P.G aym ,W .G eists,W .G engo,bm ,C.E.G erberay,Y.G ershteinaw ,2,D.G illbergf,G .G intherax,bs,

B.G �om ezh,A.G oussioucd,P.D.G rannisbt,S.G reders,H.G reenleeax,Z.D.G reenwoodbh,E.M .G regoresd,

G .G reniert,Ph.G rism ,J.-F.G rivazp,A.G rohsjeanr,S.G r�unendahlax,M .W .G r�unewaldae,F.G uobt,

J.G uobt,G .G utierrezax,P.G utierrezbw ,A.Haasbr,P.Haefnerz,S.Hagopianaw ,J.Haleybp,I.Hallbm ,

R.E.Hallau,L.Hang,K .Harderas,A.Harelbs,J.M .Hauptm anbe,J.Haysar,T.Hebbekeru,D.Hedinaz,

J.G .Hegem anai,A.P.Heinsonav,U.Heintzbj,C.Henselx,I.Heredia-DeLa Cruzah,K .Hernerbl,

G .Heskethbk,M .D.Hildrethbc,R.Hiroskycc,T.Hoangaw ,J.D.Hobbsbt,B.Hoeneisenl,M .Hohlfeldv,

S.Hossainbw ,P.Houbenai,Y.Hubt,Z.Hubacekj,N.Huskeq,V.Hynekj,I.Iashvilibq,R.Illingworthax,

A.S.Itoax,S.Jabeenbj,M .Ja�r�ep,S.Jainbw ,K .Jakobsw ,D.Jam ino,R.Jesikar,K .Johnsat,

C.Johnsonbr,M .Johnsonax,D.Johnstonbo,A.Jonckheereax,P.Jonssonar,A.Justeax,E.K ajfaszo,

D.K arm anovam ,P.A.K asperax,I.K atsanosbo,V.K aushikbz,R.K ehoeca,S.K erm icheo,N.K halatyanax,

A.K hanovbx,A.K harchilavabq,Y.N.K harzheevak,D.K hatidzeby,M .H.K irbyba,M .K irschu,B.K lim aax,

J.M .K ohliab,J.-P.K onrathw ,A.V.K ozelovan,J.K rausbm ,T.K uhly,A.K um arbq,A.K upcok,T.K ur�cat,

V.A.K uzm inam ,J.K vitai,F.Lacroixm ,D.Lam bc,S.Lam m ersbb,G .Landsbergby,P.Lebrunt,H.S.Leeaf,

W .M .Leeax,A.Le atam ,J.Lellouchq,L.Liav,Q .Z.Liax,S.M .Liettie,J.K .Lim af,D.Lincolnax,

J.Linnem annbm ,V.V.Lipaevan,R.Liptonax,Y.Liug,Z.Liuf,A.Lobodenkoao,M .Lokajicekk,P.Loveaq,

H.J.Lubatticd,R.Luna-G arciaah,4,A.L.Lyonax,A.K .A.M acielb,D.M ackincb,P.M �attigaa,

R.M aga~na-Villalbaah,P.K .M alat,S.M alikbo,V.L.M alyshevak,Y.M aravinbg,B.M artinn,

R.M cCarthybt,C.L.M cG ivernbf,M .M .M eijeraj,A.M elnitchoukbn,L.M endozah,D.M enezesaz,

P.G .M ercadantee,M .M erkinam ,K .W .M errittax,A.M eyeru,J.M eyerx,N.K .M ondalad,

H.E.M ontgom eryax,R.W .M ooref,T.M oulikbf,G .S.M uanzao,M .M ulhearnbr,O .M undalv,L.M undim c,

E.Nagyo,M .Naim uddinax,M .Narainby,H.A.Nealbl,J.P.Negreth,P.Neustroevao,H.Nilsenw ,

H.Nogim ac,S.F.Novaese,T.Nunnem annz,G .O brantao,C.O chandop,D.O noprienkobg,J.O rdunaah,

N.O shim aax,N.O sm anar,J.O stabc,R.O tecj,G .J.O tero y G arz�ona,M .O wenas,M .Padillaav,

P.Padleycb,M .Pangilinanby,N.Parasharbd,S.-J.Parkx,S.K .Parkaf,J.Parsonsbr,R.Partridgeby,

N.Paruabb,A.Patwabu,B.Penningw ,M .Per�lovam ,K .Petersas,Y.Petersas,P.P�etro�p,R.Piegaiaa,

1

J.Piperbm ,M .-A.Pleierv,P.L.M .Podesta-Lerm aah,1,V.M .Podstavkovax,Y.Pogorelovbc,M .-E.Polb,

P.Polozoval,A.V.Popovan,M .Prewittcb,S.Protopopescubu,J.Q ianbl,A.Q uadtx,B.Q uinnbn,

A.Rakitineaq,M .S.Rangelp,K .Ranjanac,P.N.Rato�aq,P.Renkelca,P.Richas,M .Rijssenbeekbt,

I.Ripp-Baudots,F.Rizatdinovabx,S.Robinsonar,M .Rom inskybw ,C.Royonr,P.Rubinovax,R.Ruchtibc,

G .Safronoval,G .Sajotn,A.S�anchez-Hern�andezah,M .P.Sandersz,B.Sanghiax,G .Savageax,L.Sawyerbh,

T.Scanlonar,D.Schailez,R.D.Scham bergerbt,Y.Scheglovao,H.Schellm anba,T.Schliephakeaa,

S.Schlobohm cd,C.Schwanenbergeras,R.Schwienhorstbm ,J.Sekaricaw ,H.Severinibw ,E.Shabalinax,

M .Sham im bg,V.Sharyr,A.A.Shchukinan,R.K .Shivpuriac,V.Siccardis,V.Sim akj,V.Sirotenkoax,

P.Skubicbw ,P.Slatterybs,D.Sm irnovbc,G .R.Snowbo,J.Snowbv,S.Snyderbu,S.S�oldner-Rem boldas,

L.Sonnenscheinu,A.Sopczakaq,M .Sosebeebz,K .Soustruzniki,B.Spurlockbz,J.Starkn,V.Stolinal,

D.A.Stoyanovaan,J.Strandbergbl,M .A.Strangbq,E.Straussbt,M .Straussbw ,R.Str�ohm erz,D.Strom ay,

L.Stutteax,S.Sum owidagdoaw ,P.Svoiskyaj,M .Takahashias,A.Tanasijczuka,W .Taylorf,B.Tillerz,

M .Titovr,V.V.Tokm eninak,I.Torchianiw ,D.Tsybychevbt,B.Tuchm ingr,C.Tullybp,P.M .Tutsbr,

R.Unalanbm ,L.Uvarovao,S.Uvarovao,S.Uzunyanaz,P.J.van den Bergai,R.Van K ootenbb,

W .M .van Leeuwenai,N.Varelasay,E.W .Varnesat,I.A.Vasilyevan,P.Verdiert,L.S.Vertogradovak,

M .Verzocchiax,M .Vesterinenas,D.Vilanovar,P.Vintar,P.Vokacj,R.W agnerbp,H.D.W ahlaw ,

M .H.L.S.W angbs,J.W archolbc,G .W attscd,M .W aynebc,G .W ebery,M .W eberax,6,L.W elty-Riegerbb,

A.W engerw ,7,M .W etsteinbi,A.W hitebz,D.W ickey,M .R.J.W illiam saq,G .W .W ilsonbf,

S.J.W im pennyav,M .W obischbh,D.R.W oodbk,T.R.W yattas,Y.Xieby,C.Xubl,S.Yacoobba,

R.Yam adaax,W .-C.Yangas,T.Yasudaax,Y.A.Yatsunenkoak,Z.Yeax,H.Ying,K .Yipbu,H.D.Yooby,

S.W .Younax,J.Yubz,C.Zeitnitzaa,S.Zelitchcc,T.Zhaocd,B.Zhoubl,J.Zhubt,M .Zielinskibs,

D.Ziem inskabb,L.Zivkovicbr,V.Zutshiaz,E.G .Zverevam

(The D � Collaboration)

aU niversidad de Buenos Aires,Buenos Aires,ArgentinabLA FEX ,Centro Brasileiro de Pesquisas F��sicas,Rio de Janeiro,Brazil

cU niversidade do Estado do Rio de Janeiro,Rio de Janeiro,BrazildU niversidade Federaldo A BC,Santo Andr�e,Brazil

eInstituto de F��sica Te�orica,U niversidade EstadualPaulista,S~ao Paulo,BrazilfU niversity ofAlberta,Edm onton,Alberta,Canada;Sim on Fraser U niversity,Burnaby,British Colum bia,Canada;York

U niversity,Toronto,O ntario,Canada and M cG illU niversity,M ontreal,Q uebec,CanadagU niversity ofScience and Technology ofChina,H efei,People’s Republic ofChina

hU niversidad de los Andes,Bogot�a,Colom biaiCenter for Particle Physics,Charles U niversity,Faculty ofM athem atics and Physics,Prague,Czech Republic

jCzech TechnicalU niversity in Prague,Prague,Czech RepublickCenter for Particle Physics,Institute ofPhysics,Academ y ofSciences ofthe Czech Republic,Prague,Czech Republic

lU niversidad San Francisco de Q uito,Q uito,Ecuadorm LPC,U niversit�e Blaise Pascal,CN R S/IN 2P3,Clerm ont,France

nLPSC,U niversit�e Joseph Fourier G renoble 1,CN R S/IN 2P3,InstitutN ationalPolytechnique de G renoble,G renoble,FranceoCPPM ,Aix-M arseille U niversit�e,CN R S/IN 2P3,M arseille,France

pLA L,U niversit�e Paris-Sud,IN 2P3/CN R S,O rsay,FranceqLPN H E,IN 2P3/CN R S,U niversit�es Paris V I and V II,Paris,France

rCEA ,Irfu,SPP,Saclay,FrancesIPH C,U niversit�e de Strasbourg,CN R S/IN 2P3,Strasbourg,France

tIPN L,U niversit�e Lyon 1,CN R S/IN 2P3,V illeurbanne,France and U niversit�e de Lyon,Lyon,FranceuIII.Physikalisches InstitutA ,RW TH Aachen U niversity,Aachen,G erm any

vPhysikalisches Institut,U niversit�atBonn,Bonn,G erm anyw Physikalisches Institut,U niversit�atFreiburg,Freiburg,G erm any

xII.Physikalisches Institut,G eorg-August-U niversit�atG �ottingen,G �ottingen,G erm anyyInstitutf�ur Physik,U niversit�atM ainz,M ainz,G erm any

zLudwig-M axim ilians-U niversit�atM �unchen,M �unchen,G erm anyaaFachbereich Physik,U niversity ofW uppertal,W uppertal,G erm any

abPanjab U niversity,Chandigarh,IndiaacD elhiU niversity,D elhi,India

adTata Institute ofFundam entalResearch,M um bai,IndiaaeU niversity College D ublin,D ublin,Ireland

afK orea D etector Laboratory,K orea U niversity,Seoul,K orea

2

agSungK yunK wan U niversity,Suwon,K oreaahCIN V ESTAV ,M exico City,M exico

aiFO M -Institute N IK H EF and U niversity ofAm sterdam /N IK H EF,Am sterdam ,The N etherlandsajRadboud U niversity N ijm egen/N IK H EF,N ijm egen,The N etherlands

akJointInstitute for N uclear Research,D ubna,RussiaalInstitute for Theoreticaland Experim entalPhysics,M oscow,Russia

am M oscow State U niversity,M oscow,RussiaanInstitute for H igh Energy Physics,Protvino,Russia

aoPetersburg N uclear Physics Institute,St. Petersburg,RussiaapStockholm U niversity,Stockholm ,Sweden,and U ppsala U niversity,U ppsala,Sweden

aqLancaster U niversity,Lancaster,U nited K ingdomarIm perialCollege,London,U nited K ingdom

asU niversity ofM anchester,M anchester,U nited K ingdomatU niversity ofArizona,Tucson,Arizona 85721,U SA

auCalifornia State U niversity,Fresno,California 93740,U SAavU niversity ofCalifornia,Riverside,California 92521,U SAaw Florida State U niversity,Tallahassee,Florida 32306,U SA

axFerm iN ationalAccelerator Laboratory,Batavia,Illinois 60510,U SAayU niversity ofIllinois atChicago,Chicago,Illinois 60607,U SA

azN orthern Illinois U niversity,D eK alb,Illinois 60115,U SAbaN orthwestern U niversity,Evanston,Illinois 60208,U SAbbIndiana U niversity,Bloom ington,Indiana 47405,U SA

bcU niversity ofN otre D am e,N otre D am e,Indiana 46556,U SAbdPurdue U niversity Calum et,H am m ond,Indiana 46323,U SA

beIowa State U niversity,Am es,Iowa 50011,U SAbfU niversity ofK ansas,Lawrence,K ansas 66045,U SA

bgK ansas State U niversity,M anhattan,K ansas 66506,U SAbhLouisiana Tech U niversity,Ruston,Louisiana 71272,U SA

biU niversity ofM aryland,College Park,M aryland 20742,U SAbjBoston U niversity,Boston,M assachusetts 02215,U SA

bkN ortheastern U niversity,Boston,M assachusetts 02115,U SAblU niversity ofM ichigan,Ann Arbor,M ichigan 48109,U SA

bm M ichigan State U niversity,EastLansing,M ichigan 48824,U SAbnU niversity ofM ississippi,U niversity,M ississippi38677,U SA

boU niversity ofN ebraska,Lincoln,N ebraska 68588,U SAbpPrinceton U niversity,Princeton,N ew Jersey 08544,U SA

bqState U niversity ofN ew York,Bu�alo,N ew York 14260,U SAbrColum bia U niversity,N ew York,N ew York 10027,U SA

bsU niversity ofRochester,Rochester,N ew York 14627,U SAbtState U niversity ofN ew York,Stony Brook,N ew York 11794,U SAbuBrookhaven N ationalLaboratory,U pton,N ew York 11973,U SA

bvLangston U niversity,Langston,O klahom a 73050,U SAbw U niversity ofO klahom a,N orm an,O klahom a 73019,U SA

bxO klahom a State U niversity,Stillwater,O klahom a 74078,U SAbyBrown U niversity,Providence,Rhode Island 02912,U SA

bzU niversity ofTexas,Arlington,Texas 76019,U SAcaSouthern M ethodistU niversity,D allas,Texas 75275,U SA

cbRice U niversity,H ouston,Texas 77005,U SAccU niversity ofV irginia,Charlottesville,V irginia 22901,U SAcdU niversity ofW ashington,Seattle,W ashington 98195,U SA

A bstract

W epresenta new m ethod form odeling thehadronicrecoilin W ! ‘� eventsproduced athadron colliders.

The recoilischosen from a library ofrecoilsin Z ! ‘‘ data eventsand overlaid on a sim ulated W ! ‘�

event. Im plem entation ofthis m ethod requires that the data recoillibrary describe the properties ofthe

m easured recoilas a function ofthe true,ratherthan the m easured,transverse m om entum ofthe boson.

3

W e addressthis issue using a m ultidim ensionalBayesian unfolding technique. W e estim ate the statistical

and system aticuncertaintiesfrom thism ethod forthe W boson m assand width m easurem entsassum ing 1

fb� 1 ofdata from theFerm ilab Tevatron.Theuncertaintiesarefound to besm alland com parableto those

ofa m oretraditionalparam eterized recoilm odel.Forthehigh precision m easurem entsthatwillbepossible

with data from Run IIofthe Ferm ilab Tevatron and from the CERN LHC,the m ethod presented in this

paper m ay be advantageous,since it does not require an understanding ofthe m easured recoilfrom �rst

principles.

Key words: W ,Z,m ass,width,hadron,collider,Tevatron,D0,recoil

PACS:12.15.Ji,13.85.Q k,14.70.Fm ,12.38.Be

1. Introduction

The W and Z bosons are m assive gauge bosons that,along with the m assless photon,m ediate elec-

troweak interactions.The predictionsfrom the standard m odel(SM )ofweak,electrom agnetic,and strong

interactions[1]fortheirm assesand widthsinclude radiativecorrectionsfrom the top quark and the Higgs

boson.W hen precision m easurem entsofthe W boson m ass(M W )arecom bined with m easurem entsofthe

top quark m assand otherelectroweak observables,lim itson the Higgsboson m asscan be extracted. The

W boson width (�W )can be directly m easured from the fraction ofW bosonsproduced athigh m ass. It

can also be inferred indirectly within the contextofthe SM from the leptonic branching fraction ofthe W

boson.Thebranching fraction,in turn,can beinferred from theratio oftheW and Z boson cross-sections

with additionaltheoreticalinputs[2].Thedirectm easurem entof�W issensitiveto vertex correctionsfrom

physicsbeyond theSM .Thecurrentworld averageforM W is80:398� 0:025 G eV [3]and thecurrentworld

averagefor�W is2:106� 0:050G eV from directm easurem ents[4].Thelargenum berofW bosonscurrently

available in data sam plescollected atthe Ferm ilab Tevatron colliderand thatwillsoon be available from

the CERN LHC collider allow m easurem ents ofM W and �W with unprecedented precision provided the

responseofthe detectorcan be m odeled with su�cientaccuracy.

In p�p and pp collisions,W and Z bosons are produced predom inantly through quark-antiquark anni-

hilation. Higher order processes can include radiated gluons or quarks that recoilagainst the boson and

introducenon-zero boson transversem om entum [5].Figure1 showsan exam plediagram fortheproduction

ofa W orZ boson with two radiated gluonsin a p�p collision.

1V isitorfrom A ugustana College,Sioux Falls,SD ,U SA .2V isitorfrom R utgers U niversity,Piscataway,N J,U SA .3V isitorfrom The U niversity ofLiverpool,Liverpool,U K .4V isitorfrom Centro de Investigacion en Com putacion -IPN ,M exico City,M exico.5V isitorfrom ECFM ,U niversidad A utonom a de Sinaloa,Culiac�an,M exico.6V isitorfrom U niversit�atBern,Bern,Switzerland.7V isitorfrom U niversit�atZ�urich,Z�urich,Switzerland.

Preprintsubm itted to N uclear Instrum ents and M ethods A July 21,2009

W e identify W and Z bosonsthrough theirleptonic decays(W ! ‘� and Z ! ‘‘ with ‘= e;�)since

thesesignatureshavelow backgrounds.Thecharged leptonscan bedetected by thecalorim eterorthem uon

system ,while the neutrino escapesundetected. W e do notreconstructparticleswhose m om entum vectors

are nearly parallelto the beam direction,and therefore we only use kinem atic variablesin the transverse

plane that is perpendicular to the beam direction. The neutrino transverse m om entum vector (~p �T ) is

inferred from them issing transverseenergy (~=ET),which can becalculated using thetransversem om enta of

the charged lepton (~p ‘T )and the recoilsystem (~uT ):

~=ET= � [~p ‘

T + ~uT ]: (1)

W e m easure ~uT by sum m ing the observed transverse energy vectorially overallcalorim etercells that are

notassociated with the reconstructed charged lepton.

The recoilsystem isdi�cultto m odelfrom �rstprinciples;unlike the decay lepton,itisa com plicated

quantity involving m any particles,as wellas e�ects related to accelerator and detector operation. The

recoilsystem is a m ixture ofthe \hard" recoilthat balances the boson transverse m om entum and \soft"

contributions,such asparticlesproduced by the spectatorquarks(the \underlying event"),otherp�p (pp)

collisionsin the sam e bunch crossing,electronicsnoise,and residualenergy in the detector from previous

bunch crossings(\pileup").Figure 2showstransverseenergiesrecordedin thecalorim eteroftheD0detector

versusazim uthalangleand pseudorapidity [6]fora typicalW ! e� candidate.Thedi�useenergy deposits

spread overm uch ofthe detectoraredueto the recoilsystem .

The various com ponents ofthis m easured recoilsystem have di�erent dependences on instantaneous

lum inosity.Forexam ple,pileup and additionalinelasticcollisionsscalewith instantaneouslum inosity,while

the contribution from the underlying event is lum inosity independent. M oreover,detector e�ects such

assuppression ofcalorim etercells with low energy to m inim ize the eventsize (zero-suppression cuts)can

introducecorrelationsbetween thecalorim eterresponseto thehard com ponentand varioussoftcom ponents

in the event,so thatthe detectorresponsesto these com ponentscannotbe m odeled independently.

Two approacheshave been used previously to m odelthe W boson event,including the recoilsystem .

O ne m ethod takesthe underlying physicsfrom a standard M onte Carlo (M C)eventgeneratorand sm ears

itparam etrically to reproducedetectore�ects[3,7,8,9].Theparam etersaretuned to an independentbut

kinem atically sim ilardata set,nam ely Z ! ‘‘events.Thesecond approach,or\Ratio M ethod",constructs

M T tem plate distributionsby directly taking Z ! ‘‘eventsfrom colliderdata,treating one ofthe leptons

asa neutrino [10].Theratio oftheZ boson m assto thecorresponding W boson m ass,taken togetherwith

the precisely m easured Z boson m ass[11]from the CERN LEP colliderdeterm inesthe W boson m ass.In

this m ethod,sm alldi�erencesin the Z and W boson line shapesand transverse m om entum and rapidity

distributionsofthe decay leptonsm ustbe taken into account.

This paper presents a novelapproach for m odeling the recoilsystem in W ! ‘� events at hadron

5

collidersthatusesrecoilsextracted directly from Z ! ‘‘colliderdata.TheZ ! ‘‘data providea m apping

between theZ boson transversem om entum (~p ZT )and thetransversem om entum oftherecoilsystem (~uT ).

Versionsofthe recoillibrary approach havebeen proposed in the past[12]thatused the m ap between the

reconstructed ~p ZT and them easured ~uT directly.In thispaperweusea two-dim ensionalBayesian unfolding

m ethod to derivea relation between the true ~p ZT and the m easured ~uT ,which allowsthe sim ulation ofthe

recoilsystem forthe sam egeneratorlevelvalue ofthe W boson transversem om entum (~pWT ).

The recoillibrary m ethod presented in this paper has m any advantages. Since the recoils are taken

directly from Z ! ‘‘ data,they re ect the event-by-event response and resolution ofthe detector. The

additionalsoftrecoilisbuiltin,asisthe com plicated zero-suppression-induced correlationsbetween itand

the hard com ponent ofthe recoil. Proper scaling ofthe recoilsystem with instantaneous lum inosity is

autom atic since the W and Z sam pleshave sim ilarinstantaneouslum inosity pro�le. The m ostsigni�cant

advantageofthism ethod liesin itssim plicity sinceitdoesnotrequirea �rst-principlesunderstanding ofthe

recoilsystem and hasno adjustable param eters. The dom inantsystem atic uncertainties ofthis approach

com efrom the lim ited statisticalpowerofthe Z ! ‘‘recoillibrary,asistrue with the otherm ethods.

In thispaper,we outline the im plem entation ofthism ethod.The m ethod istested using the W boson

m assand width m easurem ents. O nly the electron decay channelisdiscussed,butourm ethod can also be

used in the m uon decay channel. The detectorand selection criteria are described in Section 2. The M C

sim ulation sam plesused aredescribed in Section 3.W ediscussthem ethod in Section 4.In Sections5and 6,

weassesstheuncertainty on theW boson m assand width m easurem ents,and com paretheperform anceof

thism ethod with thatofa param eterized recoilm odel.The paperconcludesin Section 7.

2. T he W and Z B oson M easurem ents

W eevaluatetherecoillibrarym ethod byestim atingbiasesand statisticaland system aticuncertaintieson

theW boson m assand width m easurem entsin theelectron channel.Thetestisperform ed usingsim ulations

oftheRun IID0 detectorattheFerm ilab Tevatron,a pp colliderwith centerofm assenergyps= 1.96 TeV.

Statisticaluncertaintiesareestim ated fora data sam plecorresponding to 1 fb� 1.

2.1. The D0 Detector

TheD0 detector[13,14]consistsofa m agneticcentral-trackingsystem ,com prised ofa silicon m icrostrip

tracker (SM T) and a central�ber tracker (CFT),both located within a 2 T superconducting solenoidal

m agnet.The SM T and CFT coverj�D j< 3:0 [6]and j�D j< 1:8,respectively.

Threeuranium /liquid argon calorim etersm easureparticleenergies.Thecentralcalorim eter(CC)covers

j�D j< 1:1,and two end calorim eters (EC) extend coverage to j�D j� 4:2. In addition to the preshower

detectors,scintillatorsbetween the CC and EC cryostatsprovide sam pling ofdeveloping showersat1:1 <

6

j�D j< 1:4.TheCC issegm entedin depth intoeightlayers.The�rstfourlayersareused prim arilytom easure

theenergiesofphotonsand electronsand arecollectively called theelectrom agnetic(EM )calorim eter.The

rem aining four layers,along with the �rstfour,are used to m easure the energiesofhadrons. M ostlayers

aresegm ented into 0:1� 0:1 regionsin (�;�)[6]space.Thethird layeroftheEM calorim eterissegm ented

into 0:05� 0:05 regions.

A m uon system is located beyond the calorim etry and consists ofa layer oftracking detectors and

scintillatortriggercountersbefore1.8T irontoriods,followedbytwosim ilarlayersafterthetoroids.Tracking

atj�D j< 1 relieson 10 cm wide drifttubes,while 1 cm m ini-drifttubesareused at1< j�D j< 2.

Scintillation counters covering 2:7 < j�D j< 4:4 are used to m easure lum inosity and to indicate the

presenceofan inelastic collision in beam scrossing.

2.2. M easurem entstrategiesfor M W and �W

The W boson m assism easured from distributionsofthe following observables:the electron transverse

m om entum ~p eT ,the m issing transverseenergy

~=ET,and the transversem assM T ,given by

M T =

q

2j~p eTjj~=E

Tj[1� cos(��)]; (2)

where �� is the opening angle between ~p eT and ~=E

Tin the transverse plane. The data distributions are

com pared with probability density functions from M C sim ulationsgenerated with variousinputW boson

m ass values (\tem plates"). A binned negative log-likelihood m ethod is used to extract M W . The W

boson width ism easured using a sim ilarm ethod,exceptthatonly eventsin the high tailregion ofthe M T

distribution are used.Forthe m assm easurem ent,the �trangeswe used are [30,48]G eV forthe j~p eT jand

j~=ETjdistributions,and [60,90]G eV forthe M T distribution. Forthe width m easurem ent,we �tthe M T

distribution overthe range[100,200]G eV.

2.3. Selection criteria

A W boson candidateisidenti�ed asan isolated electrom agneticclusteraccom panied by largej~=ETj.The

electron candidate isrequired to have a showershape consistentwith thatofan electron,j~p eT j> 25 G eV,

and j�D j< 1:05.To furtherreducebackgrounds,theelectron candidateisrequired to bespatially m atched

to a reconstructed track in the centraltracking system .Additionally,we requirej~=ETj> 25 G eV,j~uT j< 15

G eV,and 50 < M T < 200 G eV.Z boson candidatesare identi�ed aseventscontaining two such electrons

with di-electron invariant m ass 70 < M ee < 110 G eV and j~uT j< 15 G eV.The selection on j~uT jhelps

to suppressbackground and to reduce the sensitivity ofthe m easurem entto uncertaintieson the detector

m odeland the theoreticaldescription ofthe pWT distribution. Since the Z sam ple hasfewereventsathigh

pZT ,the detectorand theoreticalm odelsarebestconstrained atlow boson pT .

For this analysis, both electrons from the Z boson are required to be in the centralregion of the

calorim eterbecausethe unfolding requireswell-understood detectorresolutions.

7

3. M C Sim ulation Sam ples

In thispaperweusethreedi�erentM C sim ulations.Two ofthesearefastM C sim ulationsand thethird

isa detailed fullM C sim ulation using geant [15].Thetwo fastM C sim ulationsarebuiltaround a com m on

eventgeneratorand param etricm odelforthe electron m easurem ent,butwith di�erentrecoilm odels.O ne

usesa traditionalparam eterized m ethod to m odeltherecoilsystem ,which wecall\theparam eterized recoil

m ethod".Theotherusesournew m ethod,which wecall\therecoillibrary m ethod".ThefullM C Z ! ee

sam plehastheequivalentof6.0 fb� 1 in integrated lum inosity,and thefullM C W ! e� sam plecorresponds

to 2.5 fb� 1.

Forboth fastM Cs,thepythia [16]eventgeneratorisused tosim ulatetheproduction and decayoftheW

boson,aswellasany �nalstateradiation (FSR)photons.FSR photons,ifsu�ciently closeto theelectron,

are m erged with the electron. After the event kinem atics are generated at the four-vectorlevel,detector

e�ciencies and energy response and resolution for the electron are applied. These param eterizations are

m easured using Z ! ee eventsfrom eithercolliderdata orfullM C,depending on the study.A param etric

energy dependent m odelfor resolution e�ects is used. Param eterized e�ciencies for data selection are

prepared for com paring with either data or fullM C as a function ofelectron j~p eT j,�

e,the com ponent of

the recoilalong the electron direction,the totalhadronic activity in the event,and the reconstructed z

coordinate along the beam line where the hard scattering occurred. The recoilsystem is then m odeled

eitherusing the recoillibrary orthe param eterized m odel.

Theparam eterized recoilm ethod m odelsthedetectorresponsetothehard recoilusingatwo-dim ensional

param eterization oftheresponse(both m agnitudeand direction)asestim ated using geant-sim ulated Z !

��� events. The underlying eventis m odeled using the m easured ~=ETdistribution from data taken with a

triggerthatrequiresenergy in thelum inosity m onitors(\m inim um biasevents"),and pileup and additional

interactions are m odeled using the m easured ~=ETdistributions from unsuppressed data taken on random

beam crossings(\zero biasevents"). These are com bined with the hard recoil,and data-tuned corrections

are applied to account on average for correlations between the \hard" and \soft" recoil. The correction

param eters are tuned to Z ! ee controlsam ples. The param etric m ethods ofm odeling the recoilare

furtherdiscussed in Refs.[17]and resem ble approachesused in earlierD0 and CDF m easurem entsatthe

Tevatron [3,8,9].The recoillibrary m ethod ofm odeling the recoilisdiscussed in detailin Section 4.

The geant-based M C sim ulation also uses pythia to sim ulate the production and decay ofthe W

boson,as wellasthe underlying eventand any FSR photons. These events are then propagated through

a detailed description ofthe detector. Zero bias colliderdata collected by the D0 detector with a sim ilar

instantaneouslum inosity pro�leastheW ! e� colliderdata sam pleareoverlaid on thefullM C sim ulation

to m odeladditionalcollisionsand noise in the detector. These eventsare processed through the sam e full

setofD0 reconstruction program sasdata.

8

4. T he R ecoilLibrary M ethod

4.1. Overview

The recoillibrary is built from Z ! ee events. Because the electron energies and angles are well

m easured,the m easured ~p ZT from the two electrons provides a good �rst approxim ation ofthe true ~p Z

T .

An unfolding procedureallowsthetransform ation ofthetwo-dim ensionaldistribution ofthem easured j~p ZT j

and m easured j~uT jto thatofthe true j~pZT jand m easured j~uT j. The opening angle between the m easured

~p ZT and the m easured ~uT isalso unfolded to the opening anglebetween the true ~p Z

T and the m easured ~uT

during thisprocedure.A m ap between the true j~p ZT j,the m easured j~uT j,and the scalarE T (SE T ),which

is de�ned as the scalarsum ofthe transverse energies ofallcalorim etercells exceptthose that belong to

the reconstructed electrons,isalso produced. Thism ap isnotused by the recoilm odel,butisneeded by

the electron e�ciency m odel.The �nalresultofthe recoillibrary isthe ~u T foran event,referenced to the

true ~p ZT . Thisvectorsubstitutesforthe equivalentvectorobtained in the param etrized recoilm odel. All

furthercorrectionsfore�cienciesdue to the recoilsystem are the sam e forboth the recoillibrary and the

param etrized recoilm odel.

Figure 3 showssom e exam plesofthe distribution ofthe com ponentofthe m easured recoilalong the Z

boson direction and perpendicularto the Z boson direction.

4.2. Preparing the recoillibrary

Beforeproducing a binned recoillibrary,certain event-by-eventcorrectionsm ustbeapplied to them ea-

sured recoilsystem .W e need to rem oveany biasesin the m easured recoildistribution due to the Z boson

selection requirem ents. Electron identi�cation requirem ents,forexam ple,preferentially rejecteventswith

signi�canthadronicactivity.Eventswith signi�canthadronicactivity alsohavepoorerrecoilresolution than

eventswith littlehadronicactivity.SinceZ bosonscontain two electronswhileW bosonsonly haveone,the

biaswillnotbethesam e.Theelectronsfrom Z boson decaysalsohavea higheraveragej~p eT jand a di�erent

�e distribution than electronsfrom W boson decays.To accountforthis,werem ovethe biasesfrom the Z

boson selection,and then,when a W candidate ism ade using the recoillibrary,the biasesappropriate for

a W candidate are added,asdescribed in Ref.[17]. In thissection,we describe these correctionsto the Z

boson sam ple.

4.2.1. Rem oving the two electronsfrom Z boson events

Therecoilsystem forZ ! eeeventsisde�ned astheenergiesin allcalorim etercellsexcludingthosethat

belong to the two electrons.Since the recoilsystem willin generaldepositenergy in these cells,excluding

them biasesthecom ponentoftherecoilalongtheelectron’sdirection.W ecorrectthise�ectby adding back

an approxim ation ofthe underlying energy.

9

This correction (denoted by �u ek) depends on ue

k(the projection of~uT along the electron transverse

direction),instantaneouslum inosity,and electron �e,and isestim ated using theenergiesdeposited in equiv-

alentcells thatare separated in � from the electron in W ! e� events. In addition to correcting for the

recoilenergy undertheelectron cluster,wealso correctforelectron energy thatleaksoutofthecluster.For

Z boson events,thesecorrectionsarem adeforboth electrons.In Section 5 weestim atetheuncertainty due

to these corrections.

4.2.2. M inim izing the e� ectsofFSR photons

The fullM C sim ulation indicates that roughly 6% ofthe Z ! ee events contain FSR photons with

E

T> 400 M eV that are su�ciently far from the electrons that the electron clustering algorithm at D0

does not m erge them with a reconstructed electron. These photons are thus incorrectly included in the

m easurem ent of~uT ,instead ofin ~p ZT ,resulting in a correlated bias. Since Z ! ee events contain m ore

FSR photonsthan W ! e� eventsdo,therecoillibrary builtusing Z bosonswillcontain on averagelarger

contributionsfrom FSR photons.

Ideally,these FSR photons could be rem oved from the recoil�le,and the e�ect could be separately

m odeled within the fastM C sim ulation.Since itisdi�cultto identify these FSR photonson an event-by-

eventbasis,thee�ectisreduced by raisingthelowerlim iton thee�ectivereconstructed di-electron invariant

m assto 85 G eV,reducing the fraction ofeventswith a high E T FSR photon by 25% .

Thee�ectoftherem ainingphotonsissm allbecause,foralow pT W boson,M T � 2j~p eT j+ u

ek.Therefore,

the photonswillcreate a biason the m assonly ifthey produce a biasin the com ponentof~uT parallelto

theelectron direction.W hiletheoverlaid recoilisrotated so thatthedirection ofitscorrespondingZ boson

m atches that ofthe sim ulated W boson,the directions ofthe decay electrons from Z and W are largely

uncorrelated,and the biasism ostly canceled form easurem entsusing the M T spectrum . In Section 5 the

biasdue to the FSR photonsisestim ated.

4.2.3. Correcting for electron selection e� ciencies

The selection criteria forW and Z candidatescan introduce biasesbetween the electron and the recoil

system . Since the kinem atic and geom etric properties ofW candidates are not identicalto those ofZ

candidates,they havedi�erentbiases.

Thetwo com ponentsoftheelectron selection e�ciency m odelthatm oststrongly a�ectthesebiasesare

theSE T e�ciency and theu ejje�ciency.TheSE T e�ciency describestheelectron identi�cation probability

asa function ofthe overallactivity in the detector. The ueke�ciency describesthe probability ofelectron

identi�cation as a function ofuek. This probability decreases with increasing hadronic activity along the

electron direction.

Since the recoillibrary is built from Z ! ee events,we need to rem ove the biases introduced by the

10

selection requirem entson the two electrons. W e correctforthe e�cienciesby weighting each eventin the

Z boson recoillibrary by 1=�uek

� 1=�SE T,where �ue

k

is the product ofthe ueke�ciencies and � SE T

is the

productofthe SE T e�cienciesforthe two electronsin each Z candidate.

W hen W boson eventsareproduced from a fastM C using the recoillibrary,the m ap between the true

~p ZT ,m easured ~uT ,and SE T isused to introducethebiasesappropriateforW bosonsfrom thesee�ciencies.

To sim ulate a W boson event,a random recoilischosen from the recoillibrary corresponding to the true

W boson pWT ,and a random SE T ischosen from the SE T distribution corresponding to the true W boson

pWT and thechosen recoil~uT .Theueke�ciency and SE T e�ciency arethen applied to theelectron from W

boson decays.

4.3. Unfolding m ethod

After the recoils have been corrected as discussed above,the transform ation from m easured ~p ZT and

m easured ~uT to true ~p ZT and m easured ~uT isdone using a Bayesian unfolding technique.

4.3.1. M ultidim ensionalunfolding using Bayes’s Theorem

Unfolding is a m athem atically challenging problem ,since itinvolvesthe reversalofa random process.

Becausea given truestatecan m igrateto m any m easured statesand m any di�erenttruestatescan m igrate

to thesam em easured state,wecannotunfold detectore�ectson an event-by-eventbasis.Rather,unfolding

m ethodstypically work with binned distributions.

Forthe recoillibrary m ethod,we choseto use a Bayesian unfolding approach [18].Thisapproach suits

ourneedsbecauseitisintuitive,sim pleto im plem ent,and easy to extend to them ultidim ensionalcase.The

Bayesian technique usesconditionalprobabilitiesto determ ine the probability thata given m easured state

correspondsto a particulartrue state.

Consider a distribution ofinitialstates Ii,fi= 1;2;:::;N Ig,given by P (Ii) (the probability ofevents

with initialstateIi)and a distribution of�nalstatesFj,fj= 1;2;:::;N F g,given by P (Fj)(theprobability

ofevents with �nalstate Fj),given the m easured distribution P (Fj),and the probability for each initial

stateto m igrateto each �nalstateP (FjjIi),wecan determ ine the distribution ofinitialstatesP (Ii)using

P (Ii)=

N FX

j= 1

P (IijFj)P (Fj): (3)

Using sim ulations,we can calculate P (IijFj)from P (FjjIi),the likelihood ofa truestate uctuating to

a m easured state,using Bayes’stheorem ,which is

P (AjB )=P (B jA)P (A)

P (B ): (4)

Forourparticularexam ple,with N I initialstatesand N F �nalstates,Bayes’stheorem givesus

P (IijFj)=P (FjjIi)P (Ii)

P N I

k= 1P (FjjIk)P (Ik)

: (5)

11

W e can interpret this equation as follows: the probability that a given �nalstate Fj com es from a

particularinitialstate Ii isproportionalto the probability density ofstate Ii m ultiplied by the probability

thatIi m igratesto Fj.The denom inatornorm alizesthe distribution.

O urBayesian m ethod requiresusto m akeassum ptionsregardingthedistribution ofinitialstates,P (Ii).

Although weonly useP (Ii)to calculatetheweightsused forthem easured data,thequality oftheunfolding

could depend on P (Ii).To m inim izethise�ect,them ethod isapplied iteratively,starting with a reasonable

prior for the distribution with P0(Ii), and with each successive iteration using the previous iteration’s

unfolded distribution asthenew input.Asa cross-check,them ethod isapplied with severaldi�erentinitial

P0(Ii)distributions.The iteration procedureis:

1.Choosean initialseed inputdistribution forP0(Ii).

2.Using P0(Ii) and P (FjjIi),com pute the weights P (IijFj),as derived using the Bayesian equation

shown in Eq.5.

3.Using these weights, recalculate the unfolded distribution P1(Ii) from the relationship P1(Ii) =P N F

j= 1P0(Fj)P (IijFj)described in Eq.3.

4.Repeatthe abovestepswith P1(Ii)asthe starting distribution.

5.Iterateuntilthe unfolded P (Ii)converges.

4.3.2. Unfolding the recoildistribution

Forourapplication,the recoilvectorisdescribed by the coordinates(j~uT j; t),where j~uT jisthe m ag-

nitude ofthe m easured recoiltransverse m om entum ,and t is the opening angle between the m easured

recoiland the true boson direction in the transverse plane. These recoilvectorsare stored in an array of

two-dim ensionalrecoilhistogram s(binned in j~uT jand t).Each histogram correspondsto a discretebin in

truej~p ZT jwith binsof0.25 G eV forsm allj~p Z

T j(j~p ZT j< 50 G eV)and largerbinsatlargerj~p Z

T j.

In theim plem entation ofEq.5,theinitialstateI isspeci�ed by [(j~p ZT j)ti;

tj;(j~uT j)k]and the�nalstate

F isgiven by [(j~p ZT j)sm ;

sn;(j~uT j)k],where(j~p

ZT j)t isthetrueZ boson transversem om entum ,(j~p Z

T j)s isthe

sm eared Z boson transversem om entum ,and s isthe opening anglebetween the m easured recoiland the

sm eared Z boson direction in the transverseplane.

W estartwith an initialseed distribution thatis atin (j~p ZT j)t, t,and j~uT j.W e�nd thatittakesfewer

than 10 iterationsforthe unfolding m ethod to converge. Figure 4 showsthe convergence ofthe W boson

m assand width obtained from the M T distribution,asa function ofiteration num berin fastM C studies.

The�nalvalueachieved agreeswellwith theinputvalue.Thesystem aticuncertainty on theW boson m ass

and width due to the unfolding procedureisdiscussed furtherin Section 5.

Figure5 showsan exam pledistribution oftheprobabilitiesthata Z boson with a reconstructed j~p ZT jof

7 G eV and a recoilj~uT jof3.5 G eV correspondsto di�erenttruej~p ZT jvalues.Theseprobabilitiesareused to

weightthe given recoilaswestoreitin the recoilhistogram scorresponding to the truej~p ZT j.Figures6{10

12

show variousrecoilobservablesplotted versusthe true j~p ZT j,obtained from the truth inform ation ofthese

M C sam ples,com pared with the sam e observablesplotted versusthe reconstructed j~p ZT j,before and after

theunfolding isapplied.Theunfolding correctsforaveragee�ectsofj~p ZT jsm earing on both them eansand

the RM S valuesofthese recoilobservables.

5. U ncertainties Particular to the R ecoilLibrary M ethod

To perform high statisticstestsofthe e�cacy ofthe recoillibrary,we study the m assand width values

obtained by com paring j~p eT j,j

~=ETjand M T distributionsobtained from fastM C W boson sam plescreated

using the param eterized recoilm odelwith tem plates generated from W boson sam ples created using the

recoillibrary m ethod.Therecoillibrariesaregenerated from Z ! eeeventscreated with theparam eterized

recoilm ethod.By varyingparam etersin thesim ulation used to generatetheW boson sam pleswhileleaving

the tem platesunchanged,we m easure the biasesand statisticaland system atic uncertaintieson the recoil

library m ethod forp�p collisionsatps= 1:96 TeV.The corresponding uncertaintiesforW boson m assand

width m easurem entsatthe LHC rem ain to be evaluated,butarenotexpected to be large.

5.1. Statisticalpower ofthe Z recoilsam ple

Therearesigni�cantstatisticaluncertaintiessinceweobtain therecoilsystem form odeling theW ! e�

events from the lim ited sam ple ofZ boson events. In 1 fb� 1 ofdata,after the selection cuts,we expect

approxim ately 18,000 Z ! ee events with both electrons in the centralcalorim eter,whereas in the sam e

data weexpectapproxim ately 500,000 W ! e� eventswith the electron in thecentralcalorim eter.Forthe

recoillibrary m ethod,wechooserecoilvectorsfrom thesam esetof 18,000Z ! eeeventsto m akeW boson

tem plates.O urm ethod isthuslim ited by the sizeofthe Z recoilsam ple and any statistical uctuationsit

contains. Ifwe are to rely on thism ethod asan inputto a precision m easurem ent,we need to determ ine

the extent to which the statisticallim itations ofthe Z ! ee sam ple propagate to an uncertainty on the

m easured W boson m assand width.

W eassessthestatisticaluncertaintiesoftherecoilm ethod using an ensem bleof100fastM C sim ulations

resem blingthestatisticalsituation weexpectin realdata.W egenerateW and Z boson sam plescorrespond-

ing to 1 fb� 1 ofdata using the param eterized recoilm ethod. Foreach setofW and Z boson sam ples,we

use the Z boson eventsto create a recoillibrary and then use the library to createtem platesforthe recoil

in the sim ulated W boson events.These tem platesarethen used to extractthe W boson m assand width.

Thestatisticalpowerism easured using the spread ofextracted m assesand widthsfrom these ensem bles.

Figure11showsthem easured W boson m assesand widthsfrom 100ensem blesusingtheM T distribution.

Them ean �tvalueisin good agreem entwith theinputvalue,showing thattherecoillibrary can accurately

m odelthe param eterized recoilm ethod. W e test that the recoillibrary can m odelthe fullM C events in

Section 6.

13

The statisticaluncertainty on the m ass m easurem ent due to the recoillibrary m ethod is found to be

5 M eV from the M T spectrum ,8 M eV forthe j~p eT jspectrum ,and 17 M eV forthe j~=E

Tjspectrum . These

agree with the statisticaluncertaintieson the param eterized recoilm ethod,which are found to be 6 M eV

for the M T �t,7 M eV for the j~p eT j�t,and 19 M eV for the j~=E

Tj�t. The statisticaluncertainty on the

width m easurem entdueto therecoillibrary m ethod is40 M eV using theM T spectrum and agreeswith the

statisticaluncertainty of42 M eV using the param eterized recoilm ethod.

Both the param eterized recoiland the recoillibrary m ethods use the Z boson sam ple to m odelthe

recoil. O ne m ightnaively expectthatthe additionalinform ation contained in the functionalform used in

the param eterized m ethod would give it increased statisticalpower for the sam e-sized sam ple. However,

we do notobservea lossofstatisticalpowersince the uncertaintiesfrom the two m ethodsare very sim ilar

to each other. W e have explored the reason for this by using a sim pli�ed detector m odelofW and Z

boson eventswith and withoutrecoilenergy resolution e�ectsadded,and com paring thepT -im balance(the

di�erence between j~p ZT jand the projection ofthe recoil~uT along the boson direction)distribution forthe

param eterized and library m ethods. Due to the sim ilartransverse m om entum distributionsofthe W and

Z bosons,we �nd thatthe m eansofthe pT -im balancedistribution agreewith each otherwithin statistical

uncertainty. W e also �nd that without recoilenergy resolution e�ects,there is a clearbut sm all,O (100)

M eV,increase in the RM S ofthe pT -im balance distributions for the recoillibrary m ethod,but with the

detector resolution e�ects added,the RM S ofthe pT -im balance distribution increasesto over2 G eV and

m asksany di�erencestem m ing from the di�erencebetween the param eterized recoilm ethod and therecoil

library m ethod.

5.2. System atic uncertainties

W e m entioned in Section 4 thatseverale�ects could potentially biasthe recoillibrary m ethod. These

includeunm erged FSR photons,acceptancedi�erencesbetween Z and W boson events,residuale�ciency-

related correlationsbetween the electron and the recoilsystem ,underlying energy correctionsbeneath the

electron window,and the unfolding process.The closuretestsusing fastM C described in Section 5.1 show

theoverallbiasfrom thism ethod to besm allerthan thestatisticalpowerofthetests.Nonetheless,wewant

to m akesurethatthissm all�nalbiasisnotdueto thecancellation oflargerindividualbiasesand therefore

exam ineeach e�ectindependently.

5.2.1. Unm erged FSR photons

W e m easurethe residualbiasdue to FSR photonsby �tting two setsoffastM C sim ulations,onem ade

from an unfolded,high statisticsrecoil�lewith allFSR photonsincluded and onem adefrom an equivalent

recoil�le with no FSR photons.W e �nd thatthe m assshiftbetween thesetwo sam plesis� 1 M eV forthe

M T �t,� 2 M eV forthe j~p eT j�t,and 2 M eV forthe j~=E

Tj�t,and thatthe width shiftislessthan 1 M eV.

14

5.2.2. Di� erencesin geom etric acceptance

ForW candidates,we only requirethe electron to be in the centralcalorim eter,while forZ candidates

used to create the library,we require both electronsto be in the centralcalorim eter. To testthe biasdue

to thise�ect,wegeneratetwo recoil�les.Forone recoil�le werestrictboth electronsto the centralregion

ofthe detector,aswe would in data. Forthe otherrecoil�le,we restrictonly one electron to the central

calorim eterand allow theotherelectron to beanywhere,aswith theneutrino from theW boson decay.W e

m aketem platesfrom thetwo recoil�lesand �nd thatthedi�erencesin both m easured m assand m easured

width aresm allerthan the 2 M eV statisticaluncertainty ofthisstudy.

5.2.3. E� ciency related biases

W hen we generate unfolded recoil�les,we weight the events by the reciprocals ofthe uekand SE T

e�ciencies,asdescribed in Section 4.2.3.To check ifthisapproach introducesany biases,weperform three

tests.Forone,weonly apply the ueke�ciency.In thesecond test,weonly apply theSE T e�ciency,and in

the �naltestwe apply both e�ciencies.The m axim um biasin the �tted m assorwidth overallthree tests

isused asthe system atic uncertainty. The �naluncertainty attributed to the e�ciency correctionson the

W boson m assis7 M eV fortheM T �t,7 M eV forthej~p eT j�t,and 8 M eV forthej~=E

Tj�t.Theuncertainty

ofthe W boson width isfound to be 7 M eV.

5.2.4. Uncertainty in �u ek

In Section 4.2 we observed thatby rem oving the electronsfrom the Z ! ee recoil�le,we also rem ove

the recoilenergy that underlies the electron cones. W e correctfor this e�ect by adding back the average

energy,�u ek,expected beneath theelectrons.W ethen subtracttheelectron energy thatleaksoutsideofthe

electron cone thatisincorrectly attributed to the recoilenergy.

W e assessthe system atic uncertainty due to these correctionsasfollows. Z boson recoil�lesare m ade

forthree cases:(1)no energy corrections,(2)a constantenergy correction forunderlying hadronic energy

beneath the electron cone and constant correction for the electron energy leakage,(3) the param eterized

energy correction forunderlying hadronicenergy beneath the electron coneand constantcorrection forthe

electron energy leakage.

W e then generate three setsoftem plates from each ofthese recoil�lesand m easure the shiftin �tted

W boson m assand width between these three tem plate sets. The W boson m assshiftsby 2 M eV forthe

M T �t,4 M eV forthej~p eT j�t,and 1 M eV forthej~=E

Tj�t,with a 7 M eV shiftforthewidth.W eassign the

m agnitudeofthesem axim um shiftsasthe uncertainty on thism ethod dueto the �u ekcorrection.

5.2.5. Uncertaintiesdue to im plem entation ofunfolding

Thespeci�cchoicesm adein im plem entingtheunfoldingcould introducebiasestothe�nalm easurem ent.

O urresultsm ay depend on ourchoiceofinitialdistributionsin (j~p ZT j)t, ,and j~uT j.They could alsodepend

15

on the num berofiterationsofthe unfolding procedureweapply to the recoillibrary.

W e �nd that starting with at initialdistributions in (j~p ZT j)t, ,and j~uT j,10 iterationsare su�cient

to attain convergence.W e generate the unfolded recoil�lesusing 8,10,and 12 iterationsofthe unfolding

m ethod and �nd thatthe changesin m easured m assand width extracted from M T ,j~peT j,and j~=E

Tj�tsare

within 3 M eV statisticaluncertainty ofthe fast M C study. In addition to unfolding the recoil�le using

a at initialdistribution for the recoilspectrum ,we also try severalsm oothly varying sinusoidalinitial

distributions,and �nd thatthe variation in the �nalunfolded recoil�le isnegligible.

5.3. Totalsystem atic uncertaintiesdue to the recoilsystem sim ulation

Table 1 shows the estim ated system atic uncertainties due to the recoilsystem sim ulation for 1 fb� 1

offastM C data. The overallsystem atic uncertainties,obtained by adding the individualuncertaintiesin

quadrature,arefound to be9 M eV using theM T �t,12 M eV using thej~p eT j�t,and 19 M eV using thej~=E

Tj

�tforthe W boson m ass,and 41 M eV using the M T �tforthe W boson width.

6. FullM C closure ofW boson m ass and w idth

W etestboth therecoillibrary m ethod and theparam eterized recoilm ethod using a detailed M C sam ple

produced using a geant-based fulldetectorm odelforW and Z boson production. The fullM C Z boson

sam pleisequivalentto 6.0 fb� 1 and theW boson sam pleisequivalentto 2.5 fb� 1.In thiscase,thefullM C

Z boson sam plesare used to create the recoillibrary. Tem platesare then created from W boson sam ples

m ade using the recoillibrary,and these are used to extractthe W boson m assand width. The extracted

valuesforthe W boson m assand width arecom pared to the inputvalues(closuretest).

Before �tting for the m ass and width ofthe fullM C sam ple,we test the accuracy ofthe m odelby

com paring variousfullM C distributionsto the fastM C m odelforan inputvalue ofthe W boson m assof

80.450 G eV.G ood agreem entbetween fullM C and fast M C using the recoillibrary m ethod is observed.

Figure 12 shows com parisonsbetween W ! e� fullM C and fast M C using the recoillibrary m ethod for

the M T ,j~peT j,and j~=E

Tjdistributions. The �2 between fulland fast M C sim ulations are also given and

are reasonable. The system atic uncertaintieson the electron m odel,dom inated by the uncertainty on the

electron energy scale,arefound to be 15 M eV fortheM T and j~=ETj�ts,and 12 M eV forthej~p e

T j�tforthe

W boson m ass,and 15 M eV forthe W boson width. System atic uncertaintieson the hadronic m odelare

taken from Section V.Since here we use the equivalentof6.0 fb� 1 offullM C Z ! ee recoils,we estim ate

theoveralluncertainty dueto therecoilsystem sim ulation by scaling theuncertainty dueto recoilstatistics

found in Section V by a factor of1=p6,leaving other estim ated system atic uncertainties the sam e. The

system atic uncertainty due to the recoilstatistics is 2 M eV using the M T �t,3 M eV using the j~p eT j�t,

and 7 M eV using the j~=ETj�tforthe W boson m assand 16 M eV using the M T �tforthe W boson width,

16

which agreeswith the corresponding system atic uncertainty in the param eterized recoilm odel.Taking the

system atic uncertainties estim ated in Section V,added in quadrature with these statisticaluncertainties,

we�nd thetotaluncertainty to be22 M eV fortheM T �t,24 M eV forthej~p eT j�tand 26 M eV forthej~=E

Tj

�tforthe W boson m ass,and 36 M eV forthe W boson width.

TheresultsofthefullM C m easurem entsagreewith thefullM C inputW boson m assand width values

within the uncertainties,asshown in Table 2.

7. C onclusion

W ehaveoutlined am ethod tom odelthehadronicrecoilsystem in W ! ‘� eventsusingrecoilsextracted

directly from a Z ! ‘‘data library. W e applied thism ethodology to a realistic fullM C sim ulation ofthe

D0 detector. The W boson m ass and width �ts to these M C events are in good agreem ent with the

inputparam eters,within statisticaluncertainties. They also agree with the valuesextracted using a m ore

traditionalparam eterized recoilm odel. Com parisons of sim ulated distributions using the recoillibrary

m ethod with M C givegood �2 agreem entovera fullrangeofdata observables.

This m ethod is lim ited by the statisticalpower ofthe Z boson recoilsam ple,as is the param eterized

recoilm odel.In addition tosystem atice�ectsfrom thelim ited statisticalpoweroftheZ boson sam ple,there

are severalsystem atic e�ectsdue to the im plem entation ofthe selection e�ciencies,geom etric acceptance,

the unfolding m ethod,and FSR.The uncertainty due to thesee�ectsisfound to be O (10)M eV.

Them ethod presented in thispaperhasm any advantages.Itaccurately describesthehighly com plicated

hadronicresponseand resolution forW boson recoilsin agiven calorim eter.Itincludescom plex correlations

between the hard and softcom ponentsofthe recoiland scalesthe recoilappropriately with lum inosity. It

requiresfewerassum ptions,no�rst-principlesdescription oftherecoilsystem ,and noadjustableparam eters.

Athadron colliderexperim entsatthe Run IITevatron and the LHC,thisapproach to m odeling the recoil

system iscom plem entary to the traditionalparam etricapproach.

A cknow ledgem ent

W ethank thesta�satFerm ilab and collaborating institutions,and acknowledgesupportfrom theDO E

and NSF (USA);CEA and CNRS/IN2P3 (France);FASI,Rosatom and RFBR (Russia);CNPq,FAPERJ,

FAPESP and FUNDUNESP (Brazil);DAE and DST (India);Colciencias(Colom bia);CO NACyT (M exico);

K RF and KO SEF (K orea);CO NICET and UBACyT (Argentina);FO M (TheNetherlands);STFC and the

RoyalSociety (United K ingdom );M SM T and G ACR (Czech Republic);CRC Program ,CFI,NSERC and

W estG rid Project (Canada);BM BF and DFG (G erm any);SFI(Ireland);The Swedish Research Council

(Sweden);CAS and CNSF (China);and the Alexandervon Hum boldtFoundation (G erm any).

17

R eferences

[1] S.G lashow,N ucl.Phys.B 22 (1961)579;S.W einberg,Phys.R ev.Lett.19 (1967)1264;A .Salam ,in Elem entary Particle

Theory, edited by N .Svartholm (A lm qvist and W iksells, Stockholm , 1969), p.367; W .Bardeen, H .Fritzsch, and M .

G ell-M ann,in Scale and Conform alSym m etry in H adron Physics,edited by R .G atto (W iley,N ew York,1973),p.139;

D .G rossand F.W ilczek,Phys.R ev.D 8 (1973) 3633;S.W einberg,Phys.R ev.Lett.31 (1973) 494.

[2] P.B.R enton,R ep.Prog.Phys.65 (2002) 1271.

[3] T.A altonen etal.(CD F Collaboration)Phys.R ev.Lett.99 (2007) 151801;Phys.R ev.D 77 (2008) 112001.

[4] T.A altonen etal.(CD F Collaboration),Phys.R ev.Lett.100 (2008) 071801.

[5] C.Balazs and C.-P.Yuan,Phys.R ev.D 56 (1997) 5558.

[6] D 0 usesa cylindricalcoordinate system with the z axisdirected along the beam axisin the proton direction.A ngles� and

� are the polar and azim uthalangles,respectively.Pseudorapidity is de�ned as � = � ln[tan(�=2)]where � is m easured

with respectto the interaction vertex.In the m asslesslim it,� isequivalentto the rapidity y = (1=2)ln[(E + pz)=(E � pz)].

�D isthe pseudorapidity m easured with respect to the center ofthe detector.

[7] D .G lenzinskiand U .H eintz,A nnu.R ev.N ucl.Part.Sci.50 (2000) 207;A .K otwaland J.Stark,A nnu.R ev.N ucl.Part.

Sci.58 (2008) 147.

[8] V .M .A bazov et al.(D 0 Collaboration), Phys.R ev. D 58 (1998) 092003; B. A bbott et al.(D 0 Collaboration), Phys.

R ev.D 58 (1998) 012002;B.A bbott et al.(D 0 Collaboration),Phys.R ev.D 62 (2000) 092006;V .M .A bazov et al.(D 0

Collaboration),Phys.R ev.D 66 (2002) 012001.

[9] T.A �olderet al.(CD F Collaboration),Phys.R ev.D 64 (2001) 052001.

[10] D .L.Shpakov,Ph.D .thesis,State U niversity ofN ew York atStony Brook (2000).

[11] C.A m sleret al.,Phys.Lett.B 667,1 (2008) and referencestherein.

[12] D .P.Saltzberg,Ph.D .Thesis,Enrico Ferm iInstitute,U niversity ofChicago (1994).

[13] S.A bachietal.(D 0 Collaboration),N ucl.Instrum .and M eth.A 338 (1994) 185.

[14] V .M .A bazov etal.(D 0 Collaboration),N ucl.Instrum .and M eth.A 565 (2006) 463.

[15] R .Brun and F.Carm inati,CER N Program Library Long W riteup W 5013,1993 unpublished.

[16] T.Sj}ostrand,S.M renna and P.Skands,JH EP 0605 (2006) 026.

[17] V .M .A bazov etal.(D 0 Collaboration),subm itted to Phys.R ev.Lett.i,arX iv:0908.0766.

[18] G .D ’A gostini,N ucl.Instrum .and M eth.A 362,487 (1995).

18

Pu )u (d

P

)0

(Z+

W

+e )-

(eν

g

g

p p

Figure 1:A n exam ple ofa diagram forthe production and leptonic decay ofa W /Z boson with radiated gluonsin a hadronic

collision.

19

eta

-4.7 -3

-2 -1

0 1

2 3

4.7

phi180

0

360

ET(GeV)

30

EM

ICD

HAD

CH

missing Et

Bins: 410Mean: 0.303Rms: 1.3Min: 0.00916Max: 25.7

mE_t: 25.3phi_t: 113 deg

Triggers:

Run 213825 Evt 25447557

electronneutrino

recoil

360

180

0

φ

ET(GeV)

η

0-1

12

-2

ET

0

3

-3

Figure 2:A typicalW ! e� candidate asrecorded by the D 0 detector. The two horizontalaxescorrespond to azim uthalangle

and pseudorapidity,and the verticalaxis is the transverse energy deposited at that location in the calorim eter. The energy

associated with the electron and the ~=ETthat corresponds to the neutrino are indicated. A llother energies contribute to the

m easured recoil. The longitudinalcom ponent ofthe neutrino m om entum is not determ ined,so it is displayed arbitrarily at

� = 0.

20

(GeV) Zu-20 -10 0 10 20

(G

eV)

Zu

-30

-20

-10

0

10

20

30(a)D0 MC

(GeV) Zu-20 -10 0 10 20

(G

eV)

Zu

-30

-20

-10

0

10

20

30(b)D0 MC

(GeV) Zu-20 -10 0 10 20

(G

eV)

Zu

-30

-20

-10

0

10

20

30(c)D0 MC

Figure 3:Exam plesofthe distribution ofthe com ponentofthe m easured recoilparallel(ukZ )and perpendicular(u? Z )to the

Z boson direction for three di�erent bins in true j~p Z

Tj(centered at (a) 0.4,(b) 10,and (c) 29 G eV ).Each dot represents ~uT

from a single event in the library.

distributionTIterations with M0 5 10 15 20

(G

eV)

WM

80.3

80.32

80.34

80.36

80.38

80.4

80.42(a)D0 MC

distributionTIterations with M0 5 10 15 20

(G

eV)

WΓ

1.95

2

2.05

(b)D0 MC

Figure 4:Estim ated (a)W boson m assand (b)W boson width in fastM C using the M T distribution,asa function ofnum ber

ofiterations used in the unfolding. The red line indicates the input values ofW boson m ass and width in the fast M C.The

defaultnum ber ofiterations used is10.

21

(GeV)T

True Z boson p0 2 4 6 8 10 12

Pro

bab

ility

0

0.02

0.04

0.06

0.08

D0 MC

Figure 5: The distribution ofthe probabilitiesthat a reconstructed j~p Z

Tjof7 G eV with corresponding j~uT jof3.5 G eV com es

from various true j~p Z

Tjbins.

(GeV)T

Z boson p0 5 10 15 20

> (G

eV)

T<u

4

6

8

10

12 D0 MC (a)

(GeV)T

Z boson p0 5 10 15 20

> (G

eV)

T<u

4

6

8

10

12 D0 MC (b)

Figure 6:M ean recoilj~uT jversustrue j~pZ

Tj(black �lled points)and m ean recoilj~uT jversusthe estim ate ofthe true ~p

Z

Tusing

the two electrons (red open boxes)when using (a) the two sm eared electrons directly and (b) the unfolded m ap.

22

(GeV)T

Z boson p0 5 10 15 20

> (G

eV)

Z<u

-10

-8

-6

-4

-2

0 D0 MC (a)

(GeV)T

Z boson p0 5 10 15 20

> (G

eV)

Z<u

-10

-8

-6

-4

-2

0 D0 MC (b)

Figure 7:M ean projection ofthe recoilalong the Z boson direction (< ukZ > )versustrue j~p Z

Tj(black �lled points)and m ean

projection ofthe recoilalong the boson direction versus the estim ate ofthe true ~p Z

Tusing the two electrons (red open boxes)

when using (a)the two sm eared electrons directly and (b)the unfolded m ap.

(GeV)T

Z boson p0 5 10 15 20 25 30R

MS

of

An

gle

Bet

wee

n R

eco

il an

d Z

0

0.5

1

1.5

D0 MC (a)

(GeV)T

Z boson p0 5 10 15 20 25 30R

MS

of

An

gle

Bet

wee

n R

eco

il an

d Z

0

0.5

1

1.5

D0 MC (b)

Figure 8: R M S ofthe opening angle between ~uT and ~p Z

Tversus true ~p Z

T(black �lled points) and R M S ofthe opening angle

between the recoiland the boson versusthe estim ate ofthe true j~p Z

Tjusing the two electrons (red open boxes)when using (a)

the two sm eared electrons directly and (b) the unfolded m ap.

23

(GeV)T

Z boson p0 5 10 15 20 25 30

TR

MS

of

u

2

3

4

5

D0 MC (a)

(GeV)T

Z boson p0 5 10 15 20 25 30

TR

MS

of

u

2

3

4

5

D0 MC (b)

Figure 9: R M S ofthe recoilj~uT jversus true j~pZ

Tj(black �lled points) and R M S ofthe recoilj~uT jversus the estim ate ofthe

true j~p Z

Tjusing the two electrons (red open boxes) when using (a) the two sm eared electrons directly and (b) the unfolded

m ap.

Opening angle between uT and pT

Z

0 0.982 1.963 2.945 3.927 4.909 5.890

100

200

300

400

500

Ev

en

ts/0

.4 r

ad

ian

s

D0 MC (a)

Opening angle between uT and pT

Z

0 0.982 1.963 2.945 3.927 4.909 5.890

100

200

300

400

500

Ev

en

ts/0

.4 r

ad

ian

s

D0 MC (b)

Figure 10: O pening angle between ~uT and true ~p Z

T(solid line)and opening angle between ~uT and the estim ated direction of

true ~p Z

T(points) when using (a) the two sm eared electrons directly and (b) the unfolded m ap forZ boson events with a true

j~p Z

Tjof4.0 to 4.25 G eV .

24

(GeV)WM80.38 80.4 80.42 80.44 80.46

Tes

ts/0

.004

GeV

0

5

10

15

20

25(a)D0 MC

(GeV)WΓ1.9 1.95 2 2.05 2.1 2.15 2.2

Tes

ts/0

.015

GeV

0

5

10

15 (b)D0 MC

Figure 11:(a)W boson m assand (b)width m easured in 100 ensem ble testsforeach tem plate generated from a recoil�le.The

dash line is a �t using a G aussian function. A llensem bles were generated with an input W boson m ass of80.419 G eV and

width of2.039 G eV .The�tted gaussian function forthem asshasa m ean valueof80:420� 0:001 G eV and R M S of0:005� 0:001

G eV .The values forthe width are 2:040� 0:001 G eV (m ean) and 0:040 � 0:003 (R M S) G eV .

Eve

nts

/0.5

GeV

5000

10000

15000

/ndf = 78.9/1002χ

FULL MC

FAST MC

D0 MC (a)

(GeV)TM50 60 70 80 90 100

χ

-2

0

2

Eve

nts

/0.5

GeV

10000

20000

30000/ndf = 74.5/702χ

FULL MC

FAST MC

D0 MC (b)

(GeV)T

Electron p30 40 50 60

χ

-2

0

2

Eve

nts

/0.5

GeV

5000

10000

15000

20000

25000

30000/ndf = 75.2/702χ

FULL MC

FAST MC

D0 MC (c)

Missing transverse energy (GeV)30 40 50 60

χ

-2

0

2

Eve

nts

/2 G

eV

10

210

310

410

510/ndf = 62.7/752χ

FULL MC

FAST MC

D0 MC (d)

(GeV)TM50 100 150 200

χ

-2

0

2

Figure 12: Com parison plots between fullM C (points) and fastM C produced using the recoillibrary (lines)forthe W boson

(a)M T ,(b)j~pe

Tj,(c)j~=E

Tj,and (d)M T (log scale)distributions.A lso shown are the � valuesde�ned asthe di�erence between

fullM C and fastM C yieldsdivided by the statisticaluncertainty on the fullM C yield.D i�erentrangesand bin sizesare used

for(a) and (d).

25

Table 1:Totalsystem atic uncertainties on the W boson m assand width from the recoillibrary m ethod,for1 fb� 1 ofZ boson

data.

Source �M W (M T ) �M W (j~p eT j) �M W (j~=E

Tj) �� W (M T )

(M eV) (M eV) (M eV) (M eV)

Recoilstatistics 5 8 17 40

FSR photons 1 2 2 1

E�ciency related bias 7 7 8 7

�u ek

2 4 1 7

Unfolding 3 3 3 3

System atictotal 9 12 19 41

Table 2:Finalresultofthe fullM C closure �tsforthe W boson m assand width using the recoillibrary m ethod. The fullM C

sam plesused here are equivalentto 2.5 fb� 1 ofW boson data and 6.0 fb� 1 ofZ boson data. Forthe �tted W boson m assand

width,the �rst uncertainty is statistical,the second is the system atic on the electron sim ulation,the third is the system atic

on the recoilsystem sim ulation due to Z boson statistics,and the fourth isother system atics on the recoilsystem sim ulation.

�M W representsthe di�erence between the m easured W boson m assand the inputvalue of80.450 G eV ,and �� W represents

the di�erence between the m easured W boson width and the inputvalue of2.071 G eV .

O bservable �M W (M eV)

M T 6 � 15 � 15 � 2 � 7

j~p eT j 5 � 19 � 12 � 3 � 8

j~=ETj 0 � 19 � 15 � 7 � 8

�� W (M eV)

M T � 5 � 27 � 15 � 16 � 10

26