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PublishedforSISSAbySpringerReceived:May13,2010Revised:July14,2010Accepted:September17,2010Published:October15,2010OnthenewphysicsreachofthedecaymodeBd→K0+UlrikEgede,aTobiasHurth,bJoaquimMatias,cMarcRamoncandWillReecea,daImperialCollegeLondon,LondonSW72AZ,UnitedKingdombInstituteforPhysics,JohannesGutenberg-University,D-55099Mainz,GermanycUniversitatAut`onomadeBarcelona,08193Bellaterra,Barcelona,SpaindCERN,Dept.
ofPhysics,CH-1211Geneva23,SwitzerlandE-mail:u.
egede@imperial.
ac.
uk,tobias.
hurth@cern.
ch,matias@ifae.
es,mramon@ifae.
es,will.
reece@cern.
chAbstract:WepresentacompletemethodtoconstructQCD-protectedobservablesbasedontheexclusive4-bodyB-mesondecayBd→K0+inthelowdileptonmassregion.
Thecoreofthemethodistherequirementthattheconstructedquantitiesshouldfullthesymmetriesoftheangulardistribution.
Wehaveidentiedallsymmetriesoftheangulardistributioninthelimitofmasslessleptonsandexplore:anewnon-trivialrelationbetweenthecoecientsoftheangulardistribution,thepossibilitytofullysolvethesystemfortheKamplitudes,andtheconstructionofnon-trivialobservables.
WealsopresentaphenomenologicalanalysisofthenewphysicssensitivityofangularobservablesinthedecaybasedonQCDfactorisation.
WefurtheranalysetheCP-conservingobservables,A(2)T,A(3)TandA(4)T.
TheyarepracticallyfreeoftheoreticaluncertaintiesduetothesoftformfactorsforthefullrangeofdileptonmassesratherthanjustatasinglepointasforAFB.
TheyalsohaveahighersensitivitytospecicnewphysicsscenarioscomparedtoobservablessuchasAFB.
Moreover,wecriticallyexaminethenewphysicsreachofCP-violatingobservablesviaacompleteerroranalysisduetoscaledependences,formfactorsandΛQCD/mbcorrections.
WehavedevelopedanensemblemethodtoevaluatetheerroronobservablesfromΛQCD/mbcorrections.
Finally,weexploretheexperimentalprospectsofCP-violatingobservablesandndthattheyareratherlimited.
Indeed,theCP-conserving(averaged)observablesA(i)T(withi=2,3,4)willoerabettersensitivitytolargeCPphasesandmaybemoresuitableforexperimentalanalysis.
Keywords:RareDecays,B-Physics,BeyondStandardModel,CPviolationArXivePrint:1005.
0571OpenAccessdoi:10.
1007/JHEP10(2010)056Contents1Introduction22Theoreticalframework42.
1Dierentialdecaydistribution42.
2QCDf/SCETframework62.
3EstimatingΛQCD/mbcorrections83Symmetriesandobservables93.
1Innitesimalsymmetries103.
2Explicitformofsymmetries113.
3Relationshipbetweencoecientsindierentialdistribution113.
4Experimentalissues133.
5Constructingobservables133.
6Moregeneralcases144Experimentalsensitivities154.
1Experimentalanalysis164.
1.
1Generation174.
1.
2Observablesensitivities174.
1.
3CPasymmetries174.
2Thepolynomialansatzre-examined174.
3Fitquality194.
4Discussion205AnalysisofCP-violatingobservables205.
1Preliminaries215.
2Phenomenologicalanalysis226AnalysisofCP-conservingobservables276.
1Leading-orderexpressionsofA(2)T276.
2Leading-orderexpressionsofA(5)T296.
3AnalysisofA(3)TandA(4)T317Conclusion31AKinematics32BTheoreticalinputparametersanduncertainties34–1–1IntroductionTheLHCeraisjustbeginning.
Flavourphysicswillplayanimportantcomplementaryroletodirectsearchesforthetheorythatliesbeyondthestandardmodel(SM).
Onecentralstrategyinthisperiodistoconstructobservablesthataremostlysensitivetospecictypesofnewphysics(NP),insuchawaythatadeviationcouldimmediatelyprovideinformationonthetypeofNPrequired:isospinbreakingNP,presenceofright-handedcurrents,scalars,etc.
ItisessentialtoworkinabottomupapproachinthedirectionofconstructingadecisiontreethathelpustodiscernwhichfeaturestheNPmodelmustincorporateandthentrytomatchthemintoagroupofmodels.
FewdecaysareabletoprovidesuchawealthofinformationwithdierentobservablesasBd→K0+,rangingfromforward-backwardasymmetries(AFB)andisospinasym-metriestoalargenumberofangularobservables.
EachoftheseobservablescanprovideinformationonthedierenttypesofNPmentionedabove.
FirstpublishedresultsfromBELLE[1]andBABAR[2]basedonO(100)decaysalreadydemonstratetheirfeasibility.
IntheearlyyearsofLHCrunningonewillberestrictedtothoseobservablesthatmaybeextractedfromtheangulardistributionusingrelativelysimpleanalyses.
Astudyofthoseobservablesrelevantfortherstfewfb1maybefoundin[3].
However,onceenoughstatisticshavebeenaccumulatedtoperformafullangularanalysisbasedonthefull4-bodydecaydistributionoftheBd→K0+,onehasthefreedomtodesignobservableswithreducedtheoreticaluncertaintiesandspecicNPsensitivity.
In[4],itwasproposedtoconstructobservablesthatmaximisethesensitivitytocon-tributionsdrivenbytheelectro-magneticdipoleoperatorO7,while,atthesametime,minimisingthedependenceonthepoorlyknownsoftformfactors.
Thisledtothecon-structionoftheobservableA(2)T,basedontheparallelandperpendicularspinamplitudesoftheK0.
ThebasicideabehindtheconstructionoftheobservablewasinspiredbythezeropointofAFBwhencalculatedasafunctionofthedileptonmasssquared,q2.
Thezeropointhasattractedalotofattentionbecauseofitscleanliness;onlyatthatpointonegetsacompletecancellationatLOoftheformfactordependenceanditsprecisepositionmayprovideinformationonthefundamentaltheorythatliesbeyondtheSM.
ForA(2)TthesoftformfactordependencecancelsatLO,notonlyatonepoint,butinthefullq2regionthusprovidingmuchmoreexperimentalinformation.
Moreover,theangularobservableishighlysensitivetonewright-handedcurrentsdrivenbytheoperatorO7[5],towhichAFBisblind.
Lookingforthecompletesetofangularobservablessensitivetoright-handedcurrents,oneisguidedtotheconstructionoftheso-calledA(3)TandA(4)Twhichincludelongitudinalspinamplitudes[6].
TheobservablesA(i)T(withi=2,3,4)usetheK0spinamplitudesasthefundamentalbuildingblock.
ThisprovidesmorefreedomtodisentangletheinformationonspecicWilsoncoecientsthanjustrestrictingoneselftousethecoecientsoftheangulardistributionasitwasrecentlydonein[7].
Forinstance,A(2)T,beingdirectlyproportionaltoC7enhancesitssensitivitytothetypeofNPenteringthiscoecient.
Moreover,usingeachcoecientoftheangulardistributioninsteadofselectedratiosoftheminducesalargersensitivitytothesoftformfactors.
Thespinamplitudesarenotdirectlyobservablequantities;toensurethataquantity–2–constructedoutofthespinamplitudescanbeobserved,itisnecessarythatitfullsthesamesymmetriesastheangulardistribution.
Thisobservationhastheimportantconse-quence[6]thatA(1)T(rstproposedin[8])cannotbeextractedfromtheangulardistributionbecauseitdoesnotrespectallitssymmetries.
Onlyameasurementofdenitehelicitydis-tributionswouldallowit,butthatisbeyondanyparticlephysicsexperimentthatcancurrentlybeimagined[6].
Toidentifyallthesymmetriesoftheangulardistributionisoneofthemainresultsofthispaper.
Wediscussthecountingofallthesymmetriesofthedistributionindierentscenarios,withandwithoutscalarsandwithandwithoutmassterms.
Weexplainthegeneralmethodofinnitesimaltransformationsthatallowustoidentifyallthesymmetries,andwedevelophereinfulldetailtheexplicitformofthefoursymmetriesinthemasslesscasewithnoscalars.
Asanimportantcrosscheckofthisresult,wesolveexplicitlythesetofspinamplitudesintermsofthecoecientsofthedistribution,makinguseofthreeoutofthefoursymmetries.
Twoimportantconsequencesofthisanalysisare:insolvingthesystemonenaturallyencountersanextrafreedomtoxoneofthevariables,andthereisanon-trivialconstraintbetweenthecoecientsoftheangulardistributionconsideredbeforeasfreeparameters.
Itisremarkablethatthisunexpectedconstraintisvalidforanydecaythathasthissamestructure.
Finally,weprovideanillustrativeexampleoftheuseofthemethodofdesigningobservableswithanobservablecalledA(5)Tthatmixessimultaneouslyleft/rightandper-pendicular/parallelspinamplitudesinaspecicwaythatnoneofthecoecientsoftheangulardistributionexhibits,openingdierentsensitivitiestoWilsoncoecients.
InthesecondpartofthepaperwepresentaphenomenologicalanalysisofthevariousangularobservablesbasedonaQCDfactorisation(QCDf)calculationtoNLOprecision.
Recently,averydetailedanalysisofangularquantitiesofthedecayBd→K0+invariousNPscenarios[7]andalsoananalysisoftheNPsensitivitiesofangularCPasymmetries[9]werepresented.
Incontrasttotheformerwork[7],wedonotassumethatthemainpartoftheΛQCD/mbcorrectionsareinsidetheQCDformfactors,butusethesoftformfactorsanddevelopanewensemblemethodfortreatingtheseunknowncorrectionsinasystematicway.
ThemaindierencestothelatteranalysisofCPviolatingobservablesistheredenitionoftheCPasymmetriesinordertoeliminatethesoftformfactordependenceatLOandtheinclusionoftheΛQCD/mbcorrectionsintotheerrorbudget,whichturnouttobesignicantinthepresenceofnewweakphases.
In[6]theexperimentalpreparationsforanindirectNPsearchusingtheseangularobservableswereworkedout,showingthatafullangularanalysisofthedecayBd→K0+attheLHCbexperimentoersgreatopportunities.
Were-evaluatethisanalysisinlightofthefourthsymmetryfortheangulardistributionandconcludethatithasnoeectontheestimatedexperimentalerrorsasallobservablesareindeedinvariantunderthissymmetry.
WeextendtheexperimentalsensitivitystudytoCP-violatingobservablesandshowthatevenwithanupgradedLHCbthereisnorealsensitivitytoCP-violatingNPphasesinC9andC10.
Thepaperisorganisedasfollows:section2brieyrecallthedierentialdistributioninBd→K0+andthetheoreticalframeworkofQCDfandsoft-collineareective–3–theory(SCET),section3extendsandcompletesourpreviousdiscussionaboutsymmetriesintheangulardistribution,itsexperimentalconsequencesarediscussedinsection4,andweperformaphenomenologicalanalysisoftheCP-violatingandCP-conservingobservablesinsections5and6respectively.
2TheoreticalframeworkTheseparationofNPeectsandhadronicuncertaintiesisthekeyissuewhenusingavourobservablesinaNPsearch.
OuranalysisisbasedonQCDfandSCETandcriticallyexaminestheNPreachofthoseobservablesviaadetailederroranalysisincludingtheimpactoftheunknownΛQCD/mbcorrections.
Inordertomakethepaperselfcontained,webrieyrecallthevarioustheoreticalingredientsofouranalysis.
2.
1DierentialdecaydistributionThedecayBd→K0+,withK0→Kπ+onthemassshell,iscompletelydescribedbyfourindependentkinematicvariables,thelepton-pairinvariantmasssquared,q2,andthethreeanglesθl,θK,φ.
Summingoverthespinsofthenalstateparticles,thedierentialdecaydistributionofBd→K0+canbewrittenasd4Γdq2dcosθldcosθKdφ=932πJ(q2,θl,θK,φ),(2.
1)Thedependenceonthethreeanglescanbemademoreexplicit:J(q2,θl,θK,φ)==J1ssin2θK+J1ccos2θK+(J2ssin2θK+J2ccos2θK)cos2θl+J3sin2θKsin2θlcos2φ+J4sin2θKsin2θlcosφ+J5sin2θKsinθlcosφ+(J6ssin2θK+J6ccos2θK)cosθl+J7sin2θKsinθlsinφ+J8sin2θKsin2θlsinφ+J9sin2θKsin2θlsin2φ.
(2.
2)Asthesignsoftheexpressiondependontheexactdenitionoftheangles,wehavemadetheirdenitionexplicitinappendixA.
TheJidependonproductsofthesixcomplexKspinamplitudes,AL,R,AL,R⊥andAL,R0inthecaseoftheSMwithmasslessleptons.
Eachoftheseisafunctionofq2.
Theamplitudesarejustlinearcombinationsofthewell-knownhelicityamplitudesdescribingtheB→Kπtransition:A⊥,=(H+1H1)/√2,A0=H0.
(2.
3)Twogeneralisationswillbemadefromthemasslesscasewithinouranalysis:iftheleptonsareconsideredmassivetheadditionalamplitudeAthastobeintroduced.
Andifweallowforscalaroperators,thereisanewamplitudeAS.
Bothcanbeintroducedindependently–4–oftheother.
FortheJiwendthefollowingexpressions(seealso[4,10–12]):1J1s≡a=(2+β2)4|AL⊥|2+|AL|2+(L→R)+4m2q2ReAL⊥AR⊥+ALAR,(2.
4a)J1c≡b=|AL0|2+|AR0|2+4m2q2|At|2+2Re(AL0AR0)+β2|AS|2,(2.
4b)J2s≡c=β24|AL⊥|2+|AL|2+(L→R),(2.
4c)J2c≡d=β2|AL0|2+(L→R),(2.
4d)J3≡e=12β2|AL⊥|2|AL|2+(L→R),(2.
4e)J4≡f=1√2β2Re(AL0AL)+(L→R),(2.
4f)J5≡g=√2βRe(AL0AL⊥)(L→R)mq2Re(ALAS+ARAS),(2.
4g)J6s≡h=2βRe(ALAL⊥)(L→R),(2.
4h)J6c≡h=4βmq2ReAL0AS+(L→R),(2.
4i)J7≡j=√2βIm(AL0AL)(L→R)+mq2Im(AL⊥AS+AR⊥AS),(2.
4j)J8≡k=1√2β2Im(AL0AL⊥)+(L→R),(2.
4k)J9≡m=β2Im(ALAL⊥)+(L→R),(2.
4l)withβ=14m2q2.
(2.
5)Thenotationswiththelettersa-mhasbeenincludedtomakethecomparisonto[6]easier.
NotethatJ6c=0inthemasslesscase.
TheamplitudesthemselvescanbeparametrisedintermsofthesevenB→Kformfactorsbymeansofanarrow-widthapproximation.
Theyalsodependontheshort-distanceWilsoncoecientsCicorrespondingtothevariousoperatorsoftheeectiveelectroweakHamiltonian.
Theprecisedenitionsoftheformfactorsandoftheeectiveoperatorsaregivenin[6].
AssumingonlythethreemostimportantSMoperatorsforthisdecaymode,namelyO7,O9,andO10,andthechirallyippedones,beingnumericallyrelevant,we1Thegeneralizationstothecasewhichincludesscalaroperatorswasrecentlypresentedin[7].
–5–have2AL,R⊥=N√2λ1/2(C(e)9+C(e)9)(C(e)10+C(e)10)V(q2)mB+mK++2mbq2(C(e)7+C(e)7)T1(q2),(2.
6a)AL,R=N√2(m2Bm2K)(C(e)9C(e)9)(C(e)10C(e)10)A1(q2)mBmK++2mbq2(C(e)7C(e)7)T2(q2),(2.
6b)AL,R0=N2mKq2(C(e)9C(e)9)(C(e)10C(e)10)**(m2Bm2Kq2)(mB+mK)A1(q2)λA2(q2)mB+mK++2mb(C(e)7C(e)7)(m2B+3m2Kq2)T2(q2)λm2Bm2KT3(q2),(2.
6c)At=Nλ1/2/q22(C(e)10C(e)10)A0(q2),(2.
6d)wheretheCidenotethecorrespondingWilsoncoecientsandλ=m4B+m4K+q42(m2Bm2K+m2Kq2+m2Bq2),(2.
7)N=G2Fα23·210π5m3B|VtbVts|2q2λ1/214m2q2.
(2.
8)Finallywenotethat,ifoneadditionallyconsidersscalaroperatorsthenAtismodiedbythenewWilsoncoecientsandanadditionalamplitude,AS,proportionaltotheformfactorA0(q2),isintroduced.
2.
2QCDf/SCETframeworkTheup-to-datepredictionsofexclusivemodesarebasedonQCDfanditsquantumeldtheoreticalformulation,soft-collineareectivetheory(SCET)[13,14].
Thecrucialthe-oreticalobservationisthatinthelimitwheretheinitialhadronisheavyandthenalmesonhasalargeenergy[15]thehadronicformfactorscanbeexpandedinthesmallratiosΛQCD/mbandΛQCD/E,whereEistheenergyofthemesonthatpicksupthesquarkfromtheBddecay.
Neglectingcorrectionsoforder1/mbandαs,thesevena-prioriindependentB→Kformfactorsreducetotwouniversalformfactorsξ⊥andξ[15,16].
TheserelationscanbestrictlyderivedwithintheQCDfandSCETapproachandleadtoasimplefactorisationformulaefortheB→KformfactorsFi(q2)≡Hiξ+ΦBTiΦK+O(ΛQCD/mb).
(2.
9)2Followingcommonconvention,weusetheeectiveWilsoncoecientsoftheseoperatorswhichincludecontributionsfromfour-quarkoperatorsaswell.
–6–Thereisalsoasimilarfactorisationformulaforthedecayamplitudes.
Therationaleofsuchformulaeisthatthehardvertexrenormalisations(Hi)andthehardscatteringker-nels(Ti)arequantitiesthatcanbecomputedperturbativelysotheycanbeseparatedfromthenon-perturbativefunctionsthatgowiththem;i.
e.
thelight-conewavefunctions(Φi)whichareprocess-independentandthesoftformfactors(ξ)whichenterinseveraldierentB→Kprocesses.
IngeneralwehavenomeanstocalculateΛQCD/mbcorrectionstotheQCDfamplitudessotheyaretreatedasunknowncorrections,withthemethodusedforthisdescribedinthefollowingsection.
This,ingeneral,leadstoalargeuncertaintyoftheoreticalpredictionsbasedontheQCDf/SCETwhichwewillexploresystematicallyandmakemanifestinourphenomenologicalanalysis.
Wedonotfollowheretheapproachof[7]wherethefullQCDformfactorsareusedintheQCDfformulae.
ThereitisassumedthatthemainpartoftheΛQCD/mbcorrectionsareinsidetheQCDformfactors,andadditionalΛQCD/mbcorrectionsarejustneglected.
ClearlysomeoftheΛQCD/mbcorrectionscouldbemovedintothefullQCDformfactors.
However,thereisnorobustquantitativeestimateoftheadditionalcorrectionsand,thus,itisnotallowedtoneglectthoseunknowncorrections,especiallyinviewoftheexpectedsmallnessofnewphysicseects.
Wefollowhereanotherstrategy.
Weconstructobservablesinwhichthesoftformfac-tordependencecancelsoutatleadingorder.
Thentheinuenceofthesoftformfactorstothephysicsisalmosteliminatedfromthephenomenologicalanalysisinacontrolledway.
OntheotherhandwemaketheuncertaintyduetoΛQCD/mbcorrectionsmanifestinouranalysis.
Itisnotexpectedthatthereareaslargeas2030%asintheB→ππdecayasarguedbelow.
Theinclusionofthe510%errorsduetotheΛQCD/mbcorrectionsinouranalysisisexploratoryofitsimpactonourobservables,evenattherisktobetooconservative.
Obviously,itisthisissuewhichcallsforimprovementinviewofthenewphysicsreachofthesemodes.
ThetheoreticalsimplicationsoftheQCDf/SCETapproacharerestrictedtothekine-maticregioninwhichtheenergyoftheKisoftheorderoftheheavyquarkmass,i.
e.
q2m2B.
Moreover,theinuencesofverylightresonancesbelow1GeV2questiontheQCDfresultsinthatregion.
Inaddition,thelongitudinalamplitudeintheQCDf/SCETapproachgeneratesalogarithmicdivergenceinthelimitq2→0indicatingproblemsinthetheoreticaldescriptionbelow1GeV2[13].
Thus,wewillconneouranalysisofallobservablestothedileptonmassintherange,1GeV2q26GeV2.
Usingthediscussedsimplications,theKspinamplitudesatleadingorderin1/mbandαshaveaverysimpleform:AL,R⊥=√2NmB(1s)(C(e)9+C(e)9)(C10+C10)+2mbs(C(e)7+C(e)7)ξ⊥(EK),(2.
10a)AL,R=√2NmB(1s)(C(e)9C(e)9)(C10C10)+2mbs(C(e)7C(e)7)ξ⊥(EK),(2.
10b)AL,R0=NmB2mK√s(1s)2(C(e)9C(e)9)(C10C10)+2mb(C(e)7C(e)7)ξ(EK),(2.
10c)At=NmBmK√s(1s)2C10C10ξ(EK),(2.
10d)–7–withs=q2/m2B,mi=mi/mB.
HereweneglectedtermsofO(m2K).
ThescalarspinamplitudeASisalsoproportionaltoξ(EK)inthislimit.
ThesymmetrybreakingcorrectionsoforderαscanbecalculatedintheQCDf/SCETapproach.
ThoseNLOcorrectionsareincludedinournumericalanalysisfollowing[13,14].
Theyarepresentedintheappendixof[6].
2.
3EstimatingΛQCD/mbcorrectionsOurobservableshavereducedtheoreticaluncertaintiesduetothecancellationofthesoftformfactors.
However,therelationsusedtomakethesecancellationsareonlyvalidatLOintheΛQCD/mbexpansion,andcorrectionstohigherordersareunknown.
Forthesetheoreticallycleanobservablestobeuseful,theimpactofthesecorrectionsontheobserv-ablesmustberobustlybounded.
IfNPistobediscoveredinBd→K0+,itmustbepossibletodemonstratethatanyeectseenisindeedNPandnotjusttheeectofanunknownSMcorrection.
ToevaluatetheeectoftheΛQCD/mbcorrections,weparametriseeachoftheK0spin-amplitudeswithsomeunknownlinearcorrection,Ai=Ai(1+Cieiθi),(2.
11)whereCiistherelativeamplitudeandθitherelativestrongphase.
IfwevaryCiandθiwithintheirallowedranges,anestimateforthetheoreticaluncertaintyduetotheseunknownparameterscanbefound.
Inordertomakethisparametrisationgeneric,however,extratermsmustbeintroduced.
InprincipletheeectiveHamiltonianwhichcontrolsthedecayhasthreeterms,He=H(u)SMe+H(t)SMe+H(t)NPe.
(2.
12)Thersttermisverysmallasitissuppressedbythefactorλu=VubVus/VtbVtsbutisresponsibleforalltheSMCP-violationinthedecay;thesecondtermisresponsibleforthedecayintheSM;andthethirdaddspossibleNPcontributions.
AfourthpossibletermH(u)NPegenericallydoesnotcontributetothemodelindependentamplitudesandisneglected.
EachofthesecontributionsisgeneratedbydierentsetsofdiagramsandmayhavedierentvaluesofCiandθi.
Eachamplitudemustbemodiedtoincludethethreesub-amplitudeswiththeircor-rections:A=(ASM(λu=0)ASM(λu=0))*(1+C1eiθ1)+ASM(λu=0)*(1+C2eiθ2)+(AFull(λu=0)ASM(λu=0))*(1+C3eiθ3).
(2.
13)ItisassumedthatonlyasingleNPoperatorisactivesoasnottointroduceextraterms.
Inthisformalism,theSMCP-violating,SMCP-conserving,andNPpartsoftheamplitudearethenallowedtohaveindependentΛQCD/mbcorrectionsandstrongphases.
AnestimateofthetheoreticaluncertaintyarisingfromtheunknownΛQCD/mbcorrec-tionsandstrongphasescannowbemadeusingarandomlyselectedensemble.
Foreach–8–memberoftheensemble,valuesofC13andθ13arechosenintherangesCi∈[0.
1,0.
1]orCi∈[0.
05,0.
05]andθi∈[π,π]fromarandomuniformdistribution.
Thisisdoneforthesevenamplitudes,At,AL,R0,AL,R,AL,R⊥,toprovideacompletedescriptionofthedecay.
Itisassumedthatthecorrectionsandphasesarenotfunctionsofq2,althoughinpractisetheymayactuallybe.
Anyunknowncorrelationsarealsoignored.
Whiletheseeectscouldleadtoanunderestimateofthetheoreticalenvelope,itisthoughtthatthismethodallowsforaconservativeestimateofthetheoreticaluncertaintiestobemade.
ToestimatethecontributiontothetheoreticaluncertaintiesfromΛQCD/mbcorrectionsforaparticularobservable,eachelementintheensemblewasusedtocalculatethevalueofthatobservableataxedvalueofq2.
Aoneσerrorisevaluatedastheintervalthatcontains66%ofthevaluesaroundthemedian.
ThisisdoneforbothCi∈[0.
05,0.
05]andCi∈[0.
1,0.
1]toillustratetheeectsofveandtenpercentcorrections.
Byrepeatingthisprocessfordierentvaluesofq2,bandscanbebuiltup.
NoassumptionofGaussianstatisticshasbeenmade;thebandsillustratetheprobablerangeforthetruevalueofeachobservable,giventhecurrentcentralvalue.
ThemethodallowsfortheprobabilitythatagivenexperimentalresultisduetoanunknownSMcorrectiontobefound.
Thechoice|Ci|0,AS=011174m>012084Table1.
Thedependenciesbetweenthecoecientsinthedierentialdistributionandthesym-metriesbetweentheamplitudesinseveralspecialcases.
MassiveleptonswithoutscalarsWehavethesevenamplitudesAL,R⊥,AL,R,AL,R0andAtinthiscaseandstillelevencoecients.
AsafactofelementaryquantummechanicswestillhaveaglobalphasetransformationcorrespondingtoφL=φR,buttheothertwosymmetriesfromthemasslesscasearenolongervalid.
ThereisanewsymmetryconcerningthephaseofAtgivenas:At=eiφtAt.
(3.
25)Thisleavesuswithtwosymmetrieswhereonlythedierentialformisknown.
MassiveleptonswithscalarsWenowhavealleightamplitudesand,withtheinclusionofJ6c,wehavetwelvecoecients.
Theglobalphasetransformation,φL=φR,andthephasetransformationofAtineq.
(3.
25)arestillvalid.
Inthiscase,thereisnodependencybetweenanyofthecoecients,leavinguswithtwosymmetrieswhereonlythedierentialformisknown.
Sowhileweinsomecasesonlyknowthedierentialformofthesymmetries,wearestillabletotestifobservablesrespectthesymmetries(seesection3.
5)andwecanalsodeterminetheoptimalsetofamplitudestotforinanexperimentalt(seesection3.
4).
Intable1wesummarisethefullknowledgeaboutthesymmetries.
4ExperimentalsensitivitiesIn[6],attingtechniquewasinvestigatedthatallowedtheextractionoftheK0spinamplitudesfromthefullangulardistributioninthemasslessleptonlimit.
Eq.
(2.
1)canbeinterpretedasaprobabilitydensityfunction(PDF)andnormalisednumerically.
WeparametriseitintermsofsixcomplexK0spinamplitudes,whicharefunctionsofq2only.
Inthelimitofinniteexperimentaldata,andforaxedvalueofq2,theseamplitudescanbefoundbyttingtherelativecontributionofeachangularcoecientasafunctionofthethreedecayangles.
Asdiscussedinsection3,thesymmetriesofthedistributioncanthenbeusedtoreducethenumberofunknowns;ifweconsidertherealandimaginaryamplitudecomponentsseparately,thetwelveparameterscanbereducedtoeightusingthesymmetryconstraints.
Afurtherspin-amplitudecomponentmayberemovedbynotingthatEq.
(2.
1)isonlysensitivetorelativenormalisations.
Thisleavessevenfreeparametersateachpointinq2.
However,in[6],onlythree,outoffour,symmetryconstraintswereconsidered–15–meaningthat,inprinciple,thetspresentedwereunder-constrained.
Theimplicationsofthiswillbeinvestigatedinthissection.
DespitethelargeincreasesinBd→K0+statisticsexpectedatLHCb,thenumberofsignaleventsavailablewillstillbetoosmallforaxedq2approachtobetaken.
Instead,thespin-amplitudecomponentsareparametrisedassecond-orderpolynomialsintheregionq2∈[1,6]GeV2.
ThesearenormalisedrelativetothevalueofRe(AL0)ataxedvalue,X0,ofq2.
Ratherthanttingdirectlyfortheamplitudes,weaimtoextractthecoecientsofthesepolynomials.
Thisintroducesanumberofmodelbiases:theunderlyingspinamplitudesareassumedtothesmoothlyvaryingintheq2windowconsidered.
Asnotedin[6],thiswasveriedforanumberofNPmodels.
Thereisalsoanimplicitassumptionthattheq2-dependentshapeofthespinamplitudesisinvariantunderthesymmetriesoftheangulardistribution.
Neglectingbackgroundparameters,theq2-dependentthas((124)*3)1=23freeparameterstobeextracted,or26in[6].
Thesewillbelabelledthefour-andthree-symmetrytsrespectively.
Thethree-symmetrytalthough,inprinciple,under-constrainedisabletoconvergeduetothepolynomialparametrisationemployed.
Byrequiringthatthreeofthespinamplitudecomponentsvanishforallvaluesofq2,wehaveusedourfreedomtochoosevaluesofφL,φR,andθfromeq.
(3.
10)ateachpointinq2;thevalueofθisstillfreetovary.
However,thePDF,Eq.
(2.
1),isinvariantunderchangesofθ;hence,thenegativelog-likelihood(NLL)usedduringminimisationshouldnotbesensitivetoitsvalue.
Theq2dependentshapeofeachamplitudecomponentismanifestlynotinvariantunderchangesinθ—therotationitimpliesmixestheimaginarypartsoftheleft-andright-handedamplitudes.
Thepolynomialparametrisationofthespin-amplitudecomponentsrequiresthateachamplitudemustbesmoothlyvarying.
Thetthenselectsthevalueofθforeachsignaleventwhichproducesthemostpolynomial-likedistribution,asthiswillhavethesmallestNLL.
Thegeneralminimisingalgorithmemployedisthenabletondagenuineminimumandconvergeproperly;theimpositionofthepolynomialansatzallowedtheunder-constrainedtof[6]toconvergeproperly.
Astheexperimentalobservablesareinvariantunderallfoursymmetries,theirq2dependentdistributionscanbefoundcorrectly;therearenosignicantbiasesseeninthecentralvaluesextractedcomparedtotheinputdistribution.
Smallbiasesareseenintheindividualspin-amplitudecomponents;withhindsight,correlationsbetweenthesecomponentswereinducedbythepresenceofthefourthsymmetry.
4.
1ExperimentalanalysisThediscussionaboveexplainswhythethree-symmetrytisabletoconvergesuccessfully,andsuggeststhatthereshouldbenochangeintheexperimentaluncertaintiesfoundwhentheextrasymmetryconstraintisadded.
Itisimportanttodemonstratethatthisisthecase.
Asbefore,theexperimentalsensitivitytodierentobservablescanbeestimatedusingatoyMonteCarlo(MC)approachandusedtocomparethethree-andfour-symmetryts.
–16–4.
1.
1GenerationAnensembleofdatasetsforBd→K0+canbegenerated;eachdatasetcontainsthePoisson-uctuatednumberofsignalandbackgroundeventsexpectedafterLHCbhascollected10fb1ofintegratedluminosity.
Estimatesofthesignalandbackgroundyieldsweretakenfrom[17,18]andscaledlinearly.
ThesignaldistributionwasgeneratedusingtheK0spinamplitudesdiscussedinsection2asinput.
Thecontributionfromtermsincludingthemuonmasswereincluded.
Noassumptionofpolynomialvariationoftheamplitudeswasusedinthegeneration.
ThesignalisassumedtohaveaGaussiandistributioninmBwithawidthof14MeVinawindowofmB±50MeVandaBreit-WignerinmKπwithwidth48MeVinawindowofmK0±100MeV.
Asimpliedbackgroundmodelisincluded;itisatinalldecayangles,eectivelytreatingallbackgroundascombinatorial,butfollowstheq2distributionofthesignal.
Detectoracceptanceeectsasdescribedin[17]arenottakenintoaccount.
WhenconsideringCP-conservingquantities,theBandBsamplesaresimplyconsideredtogether.
Wedonotincludeanycontributionsfromnon-resonantBd→Kπ++.
4.
1.
2ObservablesensitivitiesTheensembleofsimulateddatasetscanthenbeusedtoestimatetheexperimentalun-certaintiesexpectedforagivenintegratedluminosityatLHCb.
Foreachdataset,thefullangulartwasperformedtondthemostlikelyvalueforeachofthefreeparametersforthatdataset.
Forthethree-symmetryttherewere27freeparameters;26forthesignaldistributionandonetodescribethelevelofbackgroundseen.
Forthefour-symmetryt,only24parameterswererequired.
Intotalwecreatedanensembleof1200experimentsandwill,thus,atagivenvalueofq2,get1200dierentdeterminationsofeachobservable.
Bylookingatthepointwhere33%and47.
5%ofresultsliewithineithersideofthemedianoftheresultswecanformasymmetric1σand2σerrors.
Connectingtheseatdierentq2valuesgivesus1σand2σbandsfortheexperimentalerrorsontheobservable.
4.
1.
3CPasymmetriesThesensitivitytovariousCPasymmetrieswasalsoconsidered.
Inthiscase,separateBandBsamplesweregeneratedandtindependently.
Eachsamplehadonaveragehalfthenum-berofsignalandbackgroundeventsasthosedescribedinsection4.
1.
1.
TheresultsofaBandaBtcouldthenbecombinedbyre-normalisingtheBamplitudesfound,sothattheextractedvalueofRe(AL0)atX0wasthesameinbothsamples.
ThisgivessensitivitytoCPasymmetriesrelativetothispoint.
ByconsideringmanyBandBsamplestogether,esti-matesoftheexperimentalsensitivitytotheCPasymmetriescouldthenbefound.
Inarealmeasurement,amoresophisticatedapproachwouldbetakenwhichconsideredthetwosam-plessimultaneously;however,oursimpliedapproachgivesareasonablerstestimateoftheexperimentalsensitivitiesobtainableandallowcomparisonwiththeoreticalrequirements.
4.
2Thepolynomialansatzre-examinedAkeyassumptionofthettingapproachtakenin[6]isthatthespin-amplitudecompo-nentsaresmoothlyvaryingfunctionsintherangeq2∈[1,6]GeV2.
Itwasfoundthat–17–Figure1.
Theq2dependenceofIm(AL0)afterusingthefoursymmetriesofthefull-angulardistributiontoxIm(AL),Im(AR),Re(AL),andIm(AL⊥)tozero.
whenallfoursymmetriesofthemasslessangulardistributionaretakenintoaccount,thisassumptionnolongerholds;indeedtheshapeofthespin-amplitudecomponentsisnotinvariantunderthefoursymmetriesandtheirshapecanbedistortedsotheyarenolongerwelldescribedbysecond-orderpolynomials.
Otherparameterizationchoicesarelikelytobeequallyvulnerabletotheseproblemsunlesstheyareexplicitlyinvariantunderallsym-metriesoftheangulardistribution.
Considerthethree-symmetrycaseataxedq2value:in[6],AR0isremovedbysettingθ=arctan(AR0/AL0)oncetheirphaseshavebeenrotatedaway.
Thiscanbeunderstoodbysubstitutingthetrigonometricidentities,sin(arctan(θ))=θ√1+θ2,cos(arctan(θ))=1√1+θ2,(4.
1)intoeq.
(3.
10).
Thisintroducesa[1+(AR02/AL02)]12termintoeachnon-zeroamplitudecomponent,whichwillnotbewellbehavedasAL0→0.
Forthethree-symmetryt,theseproblemscanbeavoidedbytakingRe(AL0)asthereferenceamplitudecomponent,forcingittoberelativelylargeatX0.
However,toincludethefourthsymmetryconstraint,amorecomplicatedformmustbeusedinordertosetfouramplitudecomponentssimultaneously.
Adierentvalueofeachofthefourrotationanglesisrequiredforeverypointinq2duetothechangingspinamplitudes.
Thereisnoguaranteethatasetofrotationanglescanbefoundsuchthattheunxedspin-amplitudecomponentsresemblesmoothlyvaryingpolynomialsforallq2.
Theq2dependenceoftheSMinputamplitudeRe(AL0)isshowningure1oncethefoursymmetrieshavebeenappliedtoxIm(AL),Im(AR),Re(AL),andIm(AL⊥)tozero,asrequiredforinthenextsection.
ThisparticularfeatureiscausedbyRe(AL)→0atq2≈2GeV2;otherrotationchoicesleadtosimilarfeatures.
Thedistributioncannolongerbewelldescribedbyasecond-orderpolynomial.
Itmaybepossibletondachoiceofrotationparametersthatpreservethepolynomialfeaturesoftheinputspin-amplitudecomponents,however,therearenoguaranteethataparticularchoicewouldworkwhenfacedwithexperimentaldata.
Indeed,anincorrectchoicewillleadtobiasesinthecasewheretheparametrisationisapoormatchfortheunderlyingamplitudes.
Amoregenericsolutionisrequiredandcouldformthebasisforfurtherinvestigations.
–18–Figure2.
Thenegativelog-likelihoodfactorforthethree-symmetry(bluehatched)andfour-symmetry(redsolid)ensemblesoftsto10fb1toydatasetsofLHCbdata,assumingtheSMandwithq2∈[2.
5,6]GeV2.
Figure3.
Theglobalcorrelationfactorforthethree-symmetryandfour-symmetryensemblesoftsto10fb1toydatasetsofLHCbdata,assumingtheSMandwithq2∈[2.
5,6]GeV2.
Thecolourschemeisthesameasingure2.
4.
3FitqualityTheeectofaddingthefourthsymmetryconstraintwastested,bycomparingensem-blesofthree-andfour-symmetryts.
Thetwoensemblesweregeneratedwiththesamerandomseedvaluessothattheensembleofinputdatasetswasthesameforthetwoapproaches.
Thexedspin-amplitudecomponentswerechosentobeIm(AL),Im(AR),Re(AL),andinthecaseofthefour-symmetrytalsoIm(AL⊥).
TheamplitudeswerestillnormalisedrelativetoRe(AL)atX0=3.
5GeV2,howeverthetswereperformedintherangeq2∈[2.
5,6]GeV2toavoidthenon-polynomialfeaturesseeninthespin-amplitudecomponents,suchasshowningure1.
Thesensitivitiesfoundfortheangularobservablesarepoorerthanthosepresentedin[6],duetothedecreasedsignalstatisticsinthereducedq2window,however,itisinterestingtocomparetheperformanceofthetwottingmethods.
AhistogramoftheNLLofeachtisshowningure2.
Theensembleofthree-symmetryts(hatched)andfour-symmetryts(solid)canbeseen.
Theensembleofinputdatasetsisslightlydierentineachcaseduetoasmallnumberoffailedcomputingjobs,buttheoutputdistributionslookverysimilar.
Thisshowsthatthedepthoftheminimafoundisapproximatelythesameforthethree-andfour-symmetryts.
WecanalsointroduceaglobalcorrelationfactorGC,whichistheunsignedmeanoftheindividualglobalcorrelationcoecientscalculatedfromthefullcovariancematrix.
IttakesvaluesintherangeGC∈[0,1],wherezeroshowsallvariablesascompletelyuncorrelated,andoneshowstotaltcorrelation.
Itcanbeseeningure3thatthemeancorrelationofthetisreducedoncethefourthsymmetryistakenintoaccount.
TherearelessoutliersatverylowGCandthedistributionappearsmoreGaussian,indicatinganincreaseintstabilityhasbeenachieved.
Theconvergenceofthetstartingfromarbitraryinitialparametershasalsomuchimproved.
Figure4showstheestimatedexperimentalsensitivitiesfoundforthetheoreticallycleanobservableA(3)Tintherangeq2∈[2.
5,6]GeV2,withandwithoutthefourthsymmetry–19–Figure4.
OneandtwoσcontoursofestimatedexperimentalsensitivitytothetheoreticallycleanobservableA(3)Twithfull-angulartto10fb1ofLHCbdataassumingtheSM.
Theresultsofthethree-symmetrytareshownontheleft,andthefour-symmetrytontheright.
Thetswereperformedintherangeq2∈[2.
5,6]GeV2.
constraint.
Thetsarefor10fb1ofLHCbintegratedluminosityassumingtheSM.
Asmightbeexpectedfromgure2,thereislittledierenceintheestimatedexperimentalresolutionsseen.
Thesameconclusionisreachedwheninspectingotherobservables.
4.
4DiscussionThediscoveryofafourthsymmetryinthemasslessleptonslimitofthefull-angulardis-tributionofBd→K0+requiresthattheexperimentalanalysisproposedin[6]bere-evaluated.
Thepreviousanalysisusedthreeofthefouravailablesymmetryconstraintstoperformat,thatwas,inprinciple,under-constrained,byparametrisingtherealandimaginarypartsoftheK0spinamplitudesassecond-orderpolynomials.
Theinvarianceoftheobservablesunderallfoursymmetries,andthefreedomtotakearbitraryvaluesoftheθrotationangle,allowedthetstoconvergeandproducecorrectoutput,butintro-ducedasubtleparametrisationbias.
Astheobservablesarebydenitioninvarianttoallthesymmetries,theestimatedexperimentalsensitivitiesarethesameforthetwomethods.
Thishasbeendemonstratedinthissection.
However,theneedforthedevelopmentofanewttingmethod,sothatthefullexperimentalstatisticsavailableinq2∈[1,6]GeV2canbeused,isnowclear.
ThesensitivitiesfoundwillbesimilartothoseestimatedinRef.
[6]andinthispaperfortheCPasymmetries,butwithimprovedtstability.
5AnalysisofCP-violatingobservablesIn[10,19],itwasshownthateightCP-violatingobservablescanbeconstructedbycom-biningthedierentialdecayratesofdΓ(Bd→K0+)anddΓ(Bd→K0+).
InthissectionweanalysethetheoreticalandexperimentaluncertaintiesofthoseobservablesinordertojudgetheNPsensitivityofsuchCP-violatingobservables.
–20–5.
1PreliminariesThecorrespondingdecayratefortheCP-conjugateddecaymodeBd→K0+isgivenbyd4Γdq2dcosθldcosθKdφ=932πJ(q2,θl,θK,φ).
(5.
1)Asshownin[10],thecorrespondingfunctionsJi(q2,θl,θK,φ)areconnectedtofunctionsJiinthefollowingway:J1,2,3,4,7→J1,2,3,4,7,J5,6,8,9→J5,6,8,9,(5.
2)whereJiequalsJiwithallweakphasesconjugated.
BesidestheCPasymmetryinthedileptonmassdistribution,thereareseveralCP-violatingobservablesintheangulardistribution.
ThelatteraresensitivetoCP-violatingeectsasdierencesbetweentheangularcoecientfunctions,JiJi.
Aswasdiscussedin[10,19],andmorerecentlyin[9],thoseCPasymmetriesareallverysmallintheSM;theyoriginatefromthesmallCP-violatingimaginarypartofλu=(VubVus)/(VtbVts).
ThisweakphasepresentintheWilsoncoecientC(e)9isdoubly-CabibbosuppressedandfurthersuppressedbytheratiooftheWilsoncoecients(3C1+C2)/C9≈0.
085.
Moreover,itisimportanttonote[9,19]thattheCPasymmetriescorrespondingtoJ7,8,9areoddunderthetransformationφ→φandthus,theseasymmetriesareT-odd(Ttransformationreversesallparticlemomentaandparticlespins)whiletheotherangularCPasymmetriesareT-even.
T-oddCPasymmetriesarefavouredbecausetheyinvolvethecombinationcos(δθ)sin(δφW)ofthestrongandweakphasedierences[9,19],thus,theyarestilllargeinspiteofsmallstrongphasesaspredictedforexamplewithintheQCDF/SCETapproach.
Incontrast,T-evenCPasymmetriesinvolvethecombinationtosin(δθ)cos(δφW)[9,19].
6AnotherremarkisthattheCPasymmetriesrelatedtoJ5,6,8,9canbeextractedfrom(dΓ+dΓ)duetothepropertyeq.
(5.
2),andthuscanbedeterminedforanuntaggedequalmixtureofBandBdmesons.
ThisisimportantforthedecaymodesB0d→K0(→K0π0)+andBs→φ(→K+K)+butitislessrelevantfortheself-taggingmodeBd→K0(→K+π)+.
Recently,aQCDf/SCETanalysisoftheangularCP-violatingobservables,basedontheNLOresultsin[13,14],waspresentedforthersttime[9].
TheNLOcorrectionsareshowntobesizable.
ThecrucialimpactoftheNLOanalysisisthatthescaledependencegetsreducedtothe10%levelformostoftheCPasymmetries.
However,forsomeofthem,whichessentiallystartwithanontrivialNLOcontribution,thereisasignicantlylargerscaledependence.
Theq2-integratedSMpredictionsareallshowntobebelowthe102levelduetothesmallweakphaseasmentionedabove.
Theuncertaintiesduetotheformfactors,thescaledependence,andtheuncertaintyduetoCKMparametersareidentiedasthemainsourcesofSMerrors[9].
6WenoteherethatthisspecicbehaviourofT-oddandT-evenobservableswasshowninmanyexamplesofT-oddCPasymmetries(see[20]andreferencestherein)butageneralproofofthisstatementisstillmissingtoourknowledge.
–21–5.
2PhenomenologicalanalysisTheNPsensitivityofCP-violatingobservablesinthemodeBd→K0+wasdiscussedinamodel-independentway[9]andalsoinvariouspopularconcreteNPmodels[7].
ItwasfoundthattheNPcontributionstothephasesoftheWilsoncoecientsC7,C9,andC10andoftheirchiralcounterpartsdrasticallyenhancesuchCP-violatingobservables,whilepresentlymostofthosephasesareveryweaklyconstrained.
ItwasclaimedthattheseobservablesoercleansignalsofNPcontributions.
However,theNPreachofsuchobservablescanonlybejudgedwithacompleteanalysisofthetheoreticalandexperimentaluncertainties.
Totheverydetailedanalysesin[7,9]weaddthefollowingpoints:WeredenethevariousCPasymmetriesfollowingthegeneralmethodpresentedinourpreviouspaper[6]:anappropriatenormalisationoftheCPasymmetriesalmosteliminatesanyuncertaintiesduetothesoftformfactorswhichisoneofthemajorsourcesoferrorsintheSMprediction.
WeexploretheeectofthepossibleΛQCD/mbcorrectionsandmaketheuncertaintyduetothoseunknownΛQCD/mbcorrectionsmanifestinouranalysiswithintheSMandNPscenarios.
WeinvestigatetheexperimentalsensitivityoftheangularCPasymmetriesusingatoyMonteCarlomodelandestimatethestatisticaluncertaintyoftheobservableswithstatisticscorrespondingtoveyearsofnominalrunningatLHCb(10fb1)usingafullangulartmethod.
WediscusstheseissuesbyexampleofthetwoangularasymmetriescorrespondingtotheangularcoecientfunctionsJ6sandJ8;A6s=J6sJ6sd(Γ+Γ)/dq2,A8=J8J8d(Γ+Γ)/dq2.
(5.
3)WithintheSMtherstCPasymmetryrelatedtoJ6sturnsouttobethewell-knownforward-backwardCPasymmetrywhichwasproposedin[21,22].
AsarststepweredenethetwoCPobservables.
WemakesurethattheformfactordependencecancelsoutattheLOlevelbyusinganappropriatenormalisation:AV2s6s=J6sJ6sJ2s+J2s,AV8=J8J8J8+J8.
(5.
4)TheJiarebilinearintheKspinamplitudes,soitisclearfromtheLOformulaeeq.
(2.
6)that,followingthestrategyof[6],anyformfactordependenceatthisordercancelsoutinbothobservables.
WenotethatJ2shasthesameformfactordependenceasJ6sbuthaslargerabsolutevaluesoverthedileptonmassspectrumthatstabilisesthequantity.
Ingure5theuncertaintyduetotheformfactordependenceisestimatedinaconservativeway(seeappendixB)forA6sdenedineq.
(5.
3)andforAV6sdenedineq.
(5.
4).
Comparingtheplots,oneseesthatwiththeappropriatenormalisation,thismainsourceofhadronic–22–Figure5.
SMpredictionoftheCP-violatingobservablesA6s(left)andAV2s6s(right)asfunctionofthesquaredleptonmasswithuncertaintyduetothesoftformfactorsonly.
Noticethedierenceinscaleandthedierenceinrelativeerrorinthetwogures.
Figure6.
SMpredictionoftheCP-violatingobservablesA8(left)andAV8(right)withuncertaintyduetothesoftformfactorsonly.
Noticethedierenceinscaleofthetwogures.
uncertaintiesgetsalmosteliminated.
TheleftoveruncertaintyentersthroughtheformfactordependenceoftheNLOcontribution.
Figure6showstheanalogousresultsfortheobservableAV8.
InthesecondstepwemakethepossibleΛQCD/mbcorrectionsmanifestinournalresultsbyusingtheproceduredescribedinsection2.
3.
Itturnsoutthatinspiteofthisveryconservativeansatzforthepossiblepowercorrections,weneglectforexampleanykindofcorrelationsbetweensuchcorrectionsinthevariousspinamplitudes;theimpactofthosecorrectionsissmallerthantheSMuncertaintyincaseofthetwoobservablesAV6sandAV8.
Intheleftplotofgure7theSMerrorisgiven,includinguncertaintiesduetothescaledependenceandinputparametersandthespuriouserrorduetotheformfactors.
Intherightplottheestimatedpowercorrectionsaregiven,whichincaseoftheCP-violatingobservableAV6saresignicantlysmallerthanthecombineduncertaintyduetoscaleandinputparameters.
Figure8showsthesamefeaturefortheCP-violatingobservableAV8.
ThisresultisincontrasttotheoneforCP-averagedangularobservablesdiscussedin[6],wheretheestimatedpowercorrectionsalwaysrepresentthedominanterror.
ThereasonforthisspecicfeatureisthesmallnessoftheweakphaseintheSM.
Thus,oneexpects–23–Figure7.
SMuncertaintyinAV2s6s(left)andestimateofuncertaintyduetoΛQCD/mbcorrectionswithC1,2=10%(right).
Figure8.
SMuncertaintyinAV8(left)andestimateofuncertaintyduetoΛQCD/mbcorrections(right,lightgrey(green)correspondstoC1,2=5%,darkgrey(green)toC1,2=10%).
thattheimpactofpowercorrectionswillbesignicantlylargerwhenNPscenarioswithnewCPphasesareconsidered(seebelow).
InthethirdstepweconsidervariousNPscenarios.
Herewefollowthemodel-independentconstraintsderivedin[9]assumingonlyoneNPWilsoncoecientbeingnonzero.
Wecon-siderthreedierentNPbenchmarksscenariosofthiskind:1.
|CNP9|=2andφNP9=π8,π2,π(Red);2.
|CNP10|=1.
5andφNP10=π8,π2,π(Grey);3.
|C10|=3andφ10=π8,π2,π(Blue);wherethecoloursrefertotheonesusedinthefollowinggures.
TheabsolutevaluesoftheWilsoncoecientsarechoseninsuchawaythatthemodel-independentanalysis,assumingonenontrivialNPWilsoncoecientactingatatime,doesnotgiveanyboundonthecorrespondingNPphase.
Figure9showsourtwoobservablesinthethreescenarioswiththephasevalueπ8:theCP-violatingobservableAV6smightseparateaNPscenario(2),whilethecentralvaluesofscenarios(1)and(3)areveryclosetotheSM.
MoreoverobservableAV8seemstobesuitedtoseparatescenarios(1)and(3)fromtheSM.
–24–Figure9.
NPscenarios,assumingonenontrivialNPWilsoncoecientatatime,nexttoSMpredictionforAV2s6s(left)andAV8(right),forconcretevaluesseetext.
Figure10.
AV2s6s:EstimateofuncertaintyduetoΛQCD/mbcorrectionswithinNPscenariosasinpreviousgurewithphasesφi=π8(left)andφi=π2(right).
However,tojudgetheNPreachweneedacompleteerroranalysiswithinthethreeNPscenarios.
Asshowninsection2.
3wenowworkwiththreeweaksub-amplitudesinwhichpossiblepowercorrectionsarevariedindependently.
Theplotsingures10and11showthatthepossibleΛQCD/mbcorrectionshaveamuchlargerimpactonourtwoobservablesintheNPscenariosthanintheSMandbecomethedominatingtheoreticaluncertainty.
WealsogetsignicantlylargerpossibleΛQCD/mbcorrectionswhenchangingthevalueofthenewweakphasefromπ8toπ2.
Regardingevenlargerphasevalues,wenoteherethattheNPeectsdrasticallydecreaseagainwhenphasevaluesaroundπarechosenasexpected.
Nevertheless,inviewofthetheoreticalΛQCD/mbuncertaintiesonly,thetwoCP-violatingobservablescoulddiscriminatesomespecicNPscenarioswithnewCPphaseoforderπ8orπ2fromtheSM;incaseofAV2s6sNPscenario2,incaseofAV8NPscenario3andpossibly1.
Oneshouldalsoconsidertheadditionaltheoreticaluncertaintiesduetoscaledepen-dence,inputparametersandsoftformfactordependencieswithintheNPscenarios.
ThoseadditionaltheoreticaluncertaintiesaresizableandofthesameorderastheonesduetoΛQCD/mbcorrections:theyareshownintheleftplotsingures12and13asorangebandsoverlayingthetotalerrorsbarsincludingalsotheΛQCD/mbcorrections.
Asthelaststep,weanalysetheexperimentalsensitivityoftheangularCPasymmetries–25–Figure11.
AV8:EstimateofuncertaintyduetoΛQCD/mbcorrectionswithinNPscenariosasinpreviousgurewithphasesφi=π8(left)andφi=π2(right).
Figure12.
AV2s6s:EstimateofuncertaintyduetoΛQCD/mbcorrections(greybands)inNPscenario2,|CNP10|=1.
5andφNP10=π2withtheothertheoreticaluncertaintiesoverlaid(orangebands)andinSM(left)andexperimentaluncertainty(right).
Figure13.
AV8:EstimateofuncertaintyduetoΛQCD/mbcorrectionsinNPscenarios1(|CNP9|=2,φNP9=π2,redbands)and3(|C10|=3,φ10=π2,bluebands)withtheothertheoreticaluncertaintiesoverlaid(orangebands)andinSM(left)andexperimentaluncertainty(right).
usingatoyMonteCarlomodel.
Therightplotsingures12and13showtheestimatesofthestatisticaluncertaintyofAV6sandAV8withstatisticscorrespondingtoveyearsofnominalrunningatLHCb(10fb1).
Theinnerandouterbandscorrespondto1σand2σ–26–statisticalerrors.
TheplotsshowthatalltheNPbenchmarksarewithinthe1σrangeoftheexpectedexperimentalerrorincaseoftheobservableAV6s,andwithinthe2σrangeoftheexperimentalerrorincaseoftheobservableAV8.
Weemphasisethatfromtheexperimentalpointofviewthenormalisationisnotimportantwhencalculatingtheoverallsignicancebecausetheoverallerrorisdominatedbytheerroronthenumerator.
SotheexperimentalerroroftheobservablesA6sandA8denedineq.
(5.
3)usingthetraditionalnormalisationwillbesimilarlylargetotheoneofournewobservablesAV6sandAV8denedineq.
(5.
4).
OurnalconclusionisthatthepossibilitytodisentangledierentNPscenariosfortheCP-violatingobservablesremainsratherdicult.
FortheraredecayBd→K0+,LHCbhasnorealsensitivityforNPphasesuptovaluesofπ2(andneitheruptovaluesofπ)intheWilsoncoecientsC9,C10andtheirchiralcounterparts.
EvenSuper-LHCbwith100fb1integratedluminositydoesnotimprovethesituationsignicantly.
ThisisincontrasttotheCP-conservingobservablespresentedin[6]andfurtherdiscussedinthenextchapterwhich,bothfromthetheoreticalandexperimentalpointofview,areverypromising.
6AnalysisofCP-conservingobservablesTheCP-conservingobservablescanbeanalysedatLOinthelargerecoillimitusingtheheavy-quarkandlarge-EKexpressionsforthespinamplitudes,asrstproposedin[4].
Oneoftheadvantagesofthisapproachisthatweobtainanalyticexpressionsoftheseobservablesinaverysimpleway.
Theseexpressionscanbeusedtostudythebehaviouroftheobservableswithouthavingtorelyonnumericalcomputations,sincethemostrelevantfeaturesarisealreadyatLO.
ThemaingoalofthissectionistoperformthistypeofanalysisontheA(i)Tobservables.
6.
1Leading-orderexpressionsofA(2)TTheasymmetryA(2)T,rstproposedin[4]isgivenbyA(2)T=|A⊥|2|A|2|A⊥|2+|A|2,(6.
1)where|Ai|2=|ALi|2+|ARi|2.
Ithasasimpleform,freefromξ⊥(0)formfactordependencies,intheheavy-quark(mB→∞)andlargeK0energy(EK→∞)limits:7A(2)T=2ReC10C10+F2ReC7C7+FReC7C9|C10|2+|C10|2+F2(|C7|2+|C7|2)+|C9|2+2FRe(C7C9),(6.
2)whereF≡2mbmB/q2.
TheWilsoncoecientscantakethemostgeneralform:Ci=CSMi+|CNPi|eiφNPi,Ci=|Ci|eiφi,i=7,9,10.
(6.
3)WewillneglecthenceforwardboththetinySMweakphaseφSM9,thatarisesfromtheCKMelementsratioλu=(VubVus)/(VtbVts),andtheSMstrongphaseθSM9,smallerthan1ointhelowdileptonmassregion1GeV2q26GeV2[22].
7Noticethatalongthissectionwewilldropthesuperscript"e"thatC7andC9shouldbearinordertosimplifythenotation.
–27–Figure14.
A(2)TintheSM(green)andwithNPinC10=3eiπ8(blue),thisvalueisallowedbythemodelindependentanalysisof[9].
Theinnerlinecorrespondstothecentralvalueofeachcurve.
ThedarkorangebandssurroundingitaretheNLOresultsincludingalluncertainties(exceptforΛQCD/mb)asexplainedinthetext.
Internallightgreen/bluebands(barelyvisible)includetheestimatedΛQCD/mbuncertaintyata±5%levelandtheexternaldarkgreen/bluebandscorrespondtoa±10%correctionforeachspinamplitude.
Obviously,theobservableA(2)Tvanishesintheheavy-quarkandlargeK0energylim-itsatLOwhenalltheWilsoncoecientsaretakentobeSM-like.
Thisresultcanbeunderstoodrathereasily.
Theleft-handedstructureofweakinteractionsintheSMguar-anteesthat,intheselimits,thesquarkcreatedintheb→stransitionwillhavehelicityh(s)=1/2inthemasslesslimit(ms→0)[23].
ThissquarkwillcombinewiththespectatorquarkdoftheBdtoformtheK0mesonwithh(K0)=1or0(butnot+1),thereforeH+=0atquarklevelintheSM.
Usingeq.
(2.
3),thistranslatesintoA⊥=Aatthequarklevel,whichcorrespondstoA⊥Aatthehadronlevel[24–26].
TheNPdependenceofA(2)TcanbestudiedinamodelindependentwaybyswitchingononeWilsoncoecienteachtimeandkeepingalltheothersattheirSMvalues.
Asimpleinspectionofeq.
(6.
2)showsthatonlythechirallyippedoperatorsO7andO10giveanon-zeroexpressionforA(2)Tinourapproximation:A(2)T7=2F(FCSM7+CSM9)|C7|cos(φ7)(CSM10)2+F2|C7|2+(FCSM7+CSM9)2,(6.
4)andA(2)T10=2CSM10|C10|cos(φ10)(CSM10)2+|C10|2+(FCSM7+CSM9)2.
(6.
5)Equations.
(6.
5)and(6.
4)showthatA(2)TissensitivetoboththemodulusandthesignoftheWilsoncoecientsC7andC10.
WhenNPentersonlyC10,thefactthatC10<0intheSMmakestheobservablenegativeunlessπ2<|φ10|<π,enablingustodistinguishthesignofthisweakphase(gure14).
Likewise,ifNPappearsinC7,A(2)TwilldisplayazerointhedileptonmassspectrumwhenFCSM7+CSM9=0,whichwillcoincideexactlywiththezerooftheobservableAFBatLO[13].
AsthezeroisindependentofC7,allcurveswithCSM7shouldexhibititatq24GeV2,butifthereisalsoaNPcontributiontoC7,thezero–28–Figure15.
ObservablesA(2)TandAFBwithNPcurvesforthreeallowedcombinationsofC7andC7followingthemodelindependentanalysisof[9].
ThebandscorrespondtotheSMandthetheoreticaluncertaintyasdescribedingure14.
Thecyanline(shownwiththelabela)correspondsto(CNP7,C7)=(0.
26ei7π16,0.
2eiπ),thebrownlinebto(0.
07ei3π5,0.
3ei3π5)andthemagentalinecto(0.
03eiπ,0.
07).
willbeshiftedeithertohigherorlowervaluesofq2.
IncaseofasignipaectingC7,A(2)Twouldnothaveazeroatanyvalueofq2,exactlyasforAFB(see[27]forarecentdiscussionofdierentmechanismstoachievethis).
Infact,shouldNPenterbothO7andO7simultaneously,eq.
(6.
2)wouldimplyA(2)T7,7NP∝2F(FCSM7+CSM9)|C7|cos(φ7)+F|C7||CNP7|cos(φ7φNP7)(6.
6)whileAFB7,7NP∝FCSM7+CSM9+F|CNP7|cos(φNP7).
(6.
7)Thecomparisonofeq.
(6.
6)witheq.
(6.
7)canbeusedtoexplaintheimprovedsensitivityofA(2)TtocertaintypesofNPversusthatofAFB.
ThenumeratorofA(2)TexhibitssensitivitytotheweakphasesφNP7andφ7,havinganinterferencetermenhancedbythelargefactorF(8F48inthedileptonmassregionstudied),whileAFBisonlysensitivetoφNP7.
Thus,awiderdeparturefromtheSMbehaviouristobeexpectedinA(2)TwhenNPenterstheoperatorsO7andO7.
Thisisshowningure15usingthreedierentscenarios,describedinthecaptionofgure15,compatiblewithpresentexperimentalandtheoreticalconstraints.
Therefore,weemphasisethatA(2)TmustberegardedasanimprovedversionofAFBoncethefull-angularanalysisbecomespossible.
6.
2Leading-orderexpressionsofA(5)TIntheSM,wegetintheheavy-quarkandlarge-EKlimitsatLO:A(5)TSM=(CSM10)2+(FCSM7+CSM9)22[(CSM10)2+(FCSM7+CSM9)2],(6.
8)whichsetsthe"wave-like"behaviourofA(5)T.
Atlowq2,eq.
(6.
8)canbeusedtocheckthatA(5)T1GeV2SM0.
4.
Ontheotherhand,atthezero-pointofA(2)TandAFB,A(5)TexhibitsanabsolutemaximumofmagnitudeA(5)T4GeV2SM0.
5.
–29–Figure16.
A(5)TintheSMandwithNPinC10=3eiπ8andCNP9=2eiπ8(left)andinbothC7andC7Wilsoncoecients(right).
Thecyanline(a)correspondsto(CNP7,C7)=(0.
26ei7π16,0.
2eiπ),thebrownline(b)to(0.
07ei3π5,0.
3ei3π5)andthepinkline(d)to(0.
18eiπ2,0).
Thebandssymbolisethetheoreticaluncertaintyasdescribedingure14.
AnyinclusionofNPintheWilsoncoecientsC7,C9andC10willgiverisetotheappearanceofanextraterminthenumerator(withrespecttoeq.
(6.
8))thatwillshifttheobservablealongthey-axis.
A(5)Tπ/27NP=[(CSM10)2+F2|CNP7|2+(FCSM7+CSM9)2]2+4[FCSM10|CNP7|]22[(CSM10)2+F2|CNP7|2+(FCSM7+CSM9)2],(6.
9a)A(5)Tπ/29NP=[(CSM10)2+|CNP9|2+(FCSM7+CSM9)2]2+4[CSM10|CNP9|]22[(CSM10)2+|CNP9|2+(FCSM7+CSM9)2],(6.
9b)A(5)Tπ/210NP=[(CSM10)2|CNP10|2+(FCSM7+CSM9)2]2+4[|CNP10|(FCSM7+CSM9)]22[(CSM10)2+|CNP10|2+(FCSM7+CSM9)2].
(6.
9c)Ineq.
(6.
9)wehavechosenforsimplicitytheweakphaseφNPi=π/2fori=7,9,10,buttheyturnouttobedominatedbytheSMcontributionunlesstheNPWilsoncoecientsareverylarge.
However,iftheweakphasesassociatedtoNPWilsoncoecientsaredierentfromπ/2,theA(5)Tcurvewillgetshiftedeithertotheleftortotheright,dependingonthevalueoftheangle,asshowningure16.
NPmightalsoenterviathechirallyippedO7andO10.
ThecorrespondingLOexpressionsofA(5)Tintheheavy-quarkandhigh-EKlimitsreadA(5)T7=(CSM10)2+(FCSM7+CSM9)2F2|C7|22(CSM10)2+(FCSM7+CSM9)2+F2|C7|2(6.
10)andA(5)T10=(CSM10)2+|C10|2+(FCSM7+CSM9)22(CSM10)2+|C10|2+(FCSM7+CSM9)2.
(6.
11)–30–Equations(6.
10)and(6.
11)arebothfreefromNPweak-phasedependence.
A(5)Tevaluatedattheq2valueoftheAFBzero-pointcanbecomputedeasilyusingeq.
(6.
11),obtainingA(5)Tq20=12|(CSM10)2+|C10|2|(CSM10)2+|C10|2,(6.
12)wherethechoiceC10=0enablesustorecovertheSMpredictionA(5)T4GeV2SM=0.
5.
Ingure16(left)itcanbeseenthatfor|C10|=3thedepartureoftheNPcurveobtainedfromtheSMbehaviourisindeedlarge.
6.
3AnalysisofA(3)TandA(4)TTheobservablesA(3)TandA(4)Twererstintroducedin[6]totestthelongitudinalspinamplitudeA0inacontrolledway:A(3)T=|A0LAL+A0RAR||A0|2|A⊥|2,A(4)T=|A0LA⊥LA0RA⊥R||A0LAL+A0RAR|.
(6.
13)Unfortunately,thesimultaneousappearanceofA⊥,AandA0insidesquarerootsturnstheheavy-quarkandlarge-energylimitsintoratherawkwardexpressions,notreallyusefultoexplainthebehaviouroftheseobservablesataglance.
Therefore,weonlyoutlinetheirgeneralproperties.
Equation(6.
13)showsthatA(3)TandA(4)Tplayacomplementaryrole,asthenumeratorofA(3)TandthedenominatorofA(4)Tarethesame.
Thus,whenaminimumappearsinoneofthem,amaximumisexpectedintheotherobservableandtheotherwayaround.
Thisisindeedwhatcanbeobservedingure17.
ForthevaluesoftheWilsoncoecientschosen,NPenteringC10caneasilybedistinguishedfromtheSMcurve,displayingamaximumataround3.
5-4GeV2(exactlyintheenergyregionwhereA(4)Tisshowingaminimum),whileCNP10canonlybeclearlyidentiedusingA(4)T.
SomethingsimilarhappenswithNPenteringCSM7andC7:themodel-independentvalueschosenfortheseWilsoncoecientsdonotgiverisetoclearNPsignalsfromA(3)T,buttheycanbeeasilytoldapartusingA(4)T.
InthosesituationswheretheoriginoftheNPcurvecannotbeclearlyestablishedusingasingleobservable(forinstance,theccurveintheA(4)Tplotofgure17isverysimilartotheCNP10curve),thecombineduseofA(2)T,A(3)T,A(4)T,A(5)TandmaybeAFBenablesustoidentifywhichWilsoncoecient(s)hasacontributionfromNP.
7ConclusionInthispaperwehavepresentedhowthedecayBd→K0+canprovidedetailedknowl-edgeofNPeectsintheavoursector.
WedevelopedamethodforconstructingobservableswithspecicsensitivitytosometypesofNPwhile,atthesametime,keepingtheoreticalerrorsfromformfactorsundercontrol.
Amethodbasedoninnitesimalsymmetrieswaspresentedwhichallowsinagenericwaytoidentifyifanarbitrarycombinationofspinamplitudesisanobservableoftheangulardistribution.
Forthecaseofmasslessleptonsweidentiedtheexplicitformofallfoursymmetriespresent.
WeshowedthepossibleimpactoftheunknownΛQCD/mbcorrectionsontheNPsensitivityofthevariousangular–31–Figure17.
A(3)TandA(4)TintheSMandwithNPinandCNP10=1.
5eiπ8andC10=3eiπ8(left)andinbothC7andC7Wilsoncoecients(right).
Thecyanline(curvea)correspondsto(CNP7,C7)=(0.
26ei7π16,0.
2eiπ),thebrownline(curveb)to(0.
07ei3π5,0.
3ei3π5)andthemagentaline(curvec)to(0.
03eiπ,0.
07).
Thebandssymbolisethetheoreticaluncertaintyasdescribedingure14.
observablesinasystematicwayusinganensemblemethod.
Experimentalsensitivitytotheobservableswasevaluatedfordatasetscorrespondingto10fb1ofdataatLHCb.
Usingthesetools,wedidaphenomenologicalanalysisforbothCP-conservingandCP-violatingobservables.
TheconclusionfromthisisthattheCP-violatingobservableshaveverypoorexperimentalsensitivitywhiletheCP-conservingobservablesA(i)T(withi=2,3,4)areverypowerfulforndingNP,includingsituationswithlargeweakphases.
AcknowledgmentsJMacknowledgesnancialsupportfromFPA2005-02211,2005-SGR-00994,MRfromtheUniversitatAut`onomadeBarcelona,UEandWRfromtheScienceandTechnologyFacili-tiesCouncil(STFC),andTHfromtheEuropeannetworkHeptools.
THthankstheCERNtheorygroupforitshospitalityduringhisvisitstoCERN.
AKinematicsAssumingtheK0tobeonthemassshell,thedecayBd→K0+iscompletelyde-scribedbyfourindependentkinematicvariables;namely,thesquareofthelepton-pair–32–invariantmass,q2,andthethreeanglesθl,θKandφ.
ThesignoftheanglesfortheBdde-cayshowsgreatvariationintheliterature.
Thereforewepresenthereanexplicitdenitionofourconventionsandpointoutwherethesameordierentdenitionshavebeenused.
FirstweconsidertheBd→K0+decay.
Theangleθlistheanglebetweenthe+momentumintherestframeofthedimuonandthedirectionofthedimuonintherestframeoftheBd.
TheθKangleisinasimilarwaytheanglebetweentheKmomentumintheK0restframeandthedirectionoftheK0intherestframeoftheBd.
LetusforBd→K0+denethemomentumvectorsP+=p++p,(A.
1)Q+=p+p,(A.
2)PKπ+=pK+pπ+,(A.
3)QKπ+=pKpπ+.
(A.
4)Inthedimuonrestframe,wehavethatthe+momentumisparalleltoQ+andalsothatPKπ+pointsintheoppositedirectionofthedimuonintheBdrestframe.
Thuswecancomputetheθlangleascosθl=Q+·PKπ+|Q+||PKπ+|,(A.
5)wherethesuperscriptisusedtoindicatetheframe.
InasimilarwaywehaveintheK0restframecosθK=QKKπ+·PK+|QKKπ+||PK+|.
(A.
6)Finally,ifwegototherestframeoftheBd,wehaveφasthesignedanglebetweentheplanesdenedbythetwomuonsandtheK0decayproductsrespectively.
VectorsperpendiculartothedecayplanesareN+=PB+*QB+,NKπ+=PBKπ+*QBKπ+,(A.
7)whichletsusdeneφfromcosφ=N+·NKπ+|N+||NKπ+|,sinφ=N+*NKπ+|N+||NKπ+|·PB+|PB+|.
(A.
8)Theanglesaredenedintheintervals1cosθl1,1cosθK1,πφ<π.
(A.
9)Thedenitiongivenhereisidenticalto[6]butisdierentto[7].
However,thetwodeni-tionsresultinthesamesignsforallthecoecientsJiineq.
(2.
4).
NowfortheBd→K0+decaytheθlangleisstillspeciedwithrespecttothe+whileforθKtheangleisfortheK+.
Thisisequivalenttowhatisdonein[7].
Astheθlangledoesnotchangethesignofthelepton,wehaveJ1,2,3,4,7=J1,2,3,4,7,J5,6,8,9=J5,6,8,9.
(A.
10)–33–mB[29]5279.
50±0.
30MeVλ[29]0.
226±0.
001mK[29]896.
00±0.
25MeVA[29]0.
814±0.
022MW[29]80.
398±0.
025GeVρ[29]0.
135±0.
031MZ[29]91.
1876±0.
0021GeVη[29]0.
349±0.
017mt(mt)[13]167±5GeVΛ(nf=5)QCD[29]220±40MeVmb(mb)[30]4.
20±0.
04GeVαs(MZ)[29]0.
1176±0.
0002mc(mc)[31]1.
26±0.
02GeVαem[29]1/137fB[32]200±25MeVa⊥,1,K(2GeV)[33]0.
03±0.
03f⊥K(2GeV)[33]163±8MeVa⊥,2,K(2GeV)[33]0.
08±0.
06fK(2GeV)[33]220±5MeVmBξ(0)/(2mK)[14]0.
47±0.
09λB,+(h)[34]0.
51±0.
12GeVξ⊥(0)[7]0.
266±0.
032h[7]2.
2GeVTable2.
Summaryofinputparametersandestimateduncertainties.
inthefull-angulardistributionintheabsenceofCPviolation.
Fortheexperimentalpapers[1,2],adenitionhasbeenadoptedwhereallangulardistributionshavebeenplottedfortheBd→K0+decay,withtheBd→K0+eventsoverlaidassumingCPconservation.
InpractisethismeansthatBd→K0+eventshavethesignofcosθlreversedbeforeplotting.
Whenexperimentsprogresstomeasuringtheφangleaswell,specialcareneedstobetakentogetthedenitionscorrect.
BTheoreticalinputparametersanduncertaintiesTocomputethesoftformfactorerrorbandsingures5and6inaconservativefashion,wehaveused,asinputdata,thevaluesofξ(0)andξ⊥(0)shownintable2.
Onecannoticethattheξ⊥(0)valuehasbeentakenfrom[7],asitiscompatibletoξ⊥(0)=0.
26usedin[14],whileforξ(0)wehavekeptthevaluefrom[14]toallowforawideruncertaintyrange.
Theq2-dependenceoftheformfactorsV,A1andA2hasbeenparametrisedaccordingto[28]F(q2)=F(0)1aFq2/m2B+bFq4/m4B,(B.
1)whereF(0),aFandbFarethetparametersshownintable3of[28].
Substitutingtheoutcomesofeq.
(B.
1)into[14]ξ⊥(q2)=mBmB+mKV(q2),ξ(q2)=mB+mK2EKA1(q2)mBmKmBA2(q2),(B.
2)wecanobtainboththecentralvalueandtheassociateduncertaintycurvesforξ(q2)andξ⊥(q2)inthe1-6GeV2range.
TheseareusedtogetthettingparametersA,B,CandD–34–ofξ⊥(q2)=ξ⊥(0)1AB(q2/m2B)2,ξ(q2)=ξ(0)1CD(q2/m2B)3,(B.
3)whereA,C1withinapermilleprecision.
Thisparametrisationfollowscloselyeq.
(47)in[13]andallowsustoexploretheimpactofξ⊥(0)andξ(0)(withtheircorrespond-inguncertainties)totheCP-violatingandCP-conservingobservablesstudiedthroughoutthispaper.
Thenextstepistocomputetheamplitudes,keepingonesoftformfactorxedatthecentralvalueandvaryingtheotherintherangeallowedbyitsuncertainty.
Fromthem,theobservablescanbeobtainedinastraightforwardwayandtheerrorsaddedinquadrature.
TogeneratethetheoreticalerrorbandsnotduetoΛ/mbcorrections(plottedastheinnerorangestripsintheplotsofsections5and6)wehaveusedthecriteriaofBenekeetal.
in[13]andaddedthefollowinguncertaintiesinquadrature:therenormalisationscaleuncertaintyhasbeenfoundbyvaryingbetween2.
3and9.
2GeV(whereisthescaleatwhichtheWilsonCoecients,αsandtheMSmassesareevaluated),theuncertaintyintheratiomc/mbbyvaryingthisquantitybetween0.
29and0.
31,andtheotherparametricuncertaintieshavebeencollectedintothefactor[6]κ(q2)=π2fBfK,z()NcmBξz(q2)withz=⊥,(B.
4)thatdeterminestherelativemagnitudeofthehard-scatteringversustheformfactorterm[13],whichisuncertainbyabout±35%.
InournumericalanalysiswehaveusedthevaluesoftheWilsoncoecientsintable1ofref.
[13].
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