BEAMDYNAMICSCHARACTERIZATIONANDUNCERTAINTIESIN THEMUON g -2EXPERIMENTATFERMILAB By DavidAlbertoTarazona ADISSERTATION Submittedto MichiganStateUniversity inpartialentoftherequirements forthedegreeof Physics|DoctorofPhilosophy 2021 ABSTRACT BEAMDYNAMICSCHARACTERIZATIONANDUNCERTAINTIESINTHEMUON g -2EXPERIMENTATFERMILAB By DavidAlbertoTarazona Themeasurementofthepositivemuonmagneticanomaly, a ( g 2) = 2,from theFermiNationalAcceleratorLaboratory(Fermilab)Muon g -2Experiment(E989)yielded anexperimentalrelativeuncertaintyof0 : 46ppm,whichcombinedwiththepreviousmea- surementfromtheBrookhavenNationalLaboratory(BNL)Muon g -2Experiment(E821) fromthecurrentStandardModel(SM)predictionby4 : 2standarddeviations.In contrasttoE821,thegoaloftheexperimentatFermilabistodeliverameasurementof theanomalytoaprecisionof0 : 14ppmorlessinordertoreachmorethan5 ˙ discrepancy withtheSMand,therefore,stronglyestablishevidencefornewphysics.Inviewofthis stringentdetermination,athoroughdescriptionofthedelivery,storage,anddynamicsof thedetectedmuonbeamsetsthestageforconstrainingbeam-dynamicsdriventothe muonmagneticanomalyattheppblevel.Tothatextent,thisdissertationintroducesthe background,principles,andbeamrequirementsofE989;elaboratesdata-drivennumerical modelsoftheBeamDeliverySystemandMuon g -2StorageRingatFermilab;characterizes thelinearandnonlineardynamicsofthemuonbeaminthestoragering;anddescribesthe contributionstothequanofthelargestbeam-dynamicssystematiccorrectionsand theiruncertaintiesintheexperimentderivedfromthiswork. Copyrightby DAVIDALBERTOTARAZONA 2021 ToFatherandMother. iv ACKNOWLEDGMENTS Thefollowingdocumentandtheworkthereinelaboratedarenottheoutcomesofan individualInstead,theybeararesemblancetoasong,whereIhappenedtobethe soloistofthissppiecewhilebeingpartofanorchestramadeofoutstandinginterpreters andpassionatedirectors.Iamdeeplythankfultoeachofthemforalltheirhelp. TheprimarydirectorsweremyresearchadvisorsMartinBerzandMikeSyphers.Iam deeplyindebtedtoMartinforthescienguidanceovertheyearsand,especially,for teachingmehowtobebraveagainstchallengesthatseemedimpossibletoresolve.Iam profoundlygratefultoMikeforhavingintroducedmetotheofbeamphysics,forallhis brilliantlessonsandsupport,andforbeingaconstantinspirationofexcellenceinboththe scienandhumanspheres. Simultaneously,closeinteractionswithBillMorseandJamesMottallowedmetomove forwardwithintheMuon g -2Collaboration.ToBill,Itakethisopportunitytothankhim forallhisencouragementfromtheverybeginning,forthegoldenideasthatcontributedto thiswork,andforconstantlysharinghis optics :\Natureissmarterthanus,sowehaveto behumble!"ToJames,manythanksforsharinggreatideasandskills;ithasbeenafortune tolearnfromoneofthebest. IalsothankKyokoMakinoforsharingherremarkablelevelofdetailandaccuracy, whichcontributedtothiswork,andallthemembersofmyPh.D.guidancecommittee: WadeFisher, OscarNaviliatCuncic,andChong-YuRuan,togetherwithMartin,Mike,and Kyoko.IparticularlythankRemcoZegers,thegraduateprogramdirectoroftheDepartment ofPhysicsandAstronomy,forhisstrongsupportduringtheproblematicCOVIDeraand KimCrosslanatMSUforherconstantembracement.Manythankstomygraduateschool v fellowsatMSU:MaxwellCao,KirillMoskovtsev,EremeyValetov,andAdrianWeisskopf.I wasluckytohavesharedmytimewiththemoverthelastyears. ManyscientistsandengineersoftheMuon g -2CollaborationatFermilabmadethework describedinthisdissertationpossible.TheTracker,Field,Quad,Kicker,BeamDynamics, andPhaseAcceptanceteamsprovidedessentialsupportandexperimentaldatafortheanal- ysispresentedinthisdocument.Inparticular,mygratitudegoestoEliaBottalico,Jason Crnkovic,ReneeFatemi,AlejandroGarcJoeGrange,DaveHertzog,BrynnMacCoy,Joe Price,DiktysStratakis,ErikSwanson,andVolodyaTishchenko. IamobligedtoBobZwaskaandthePh.D.AcceleratorProgramatFermilabforthe generoussupportoverthelastyears,whichallowedmetheprivilegeofhavingspentat Fermilabtheportionofthisexperienceinclosecontactwiththeexperimentandsci- entiststherein.Thankyou,SusanWinchesterandtheUSParticleAcceleratorSchool,for alltheopportunitiesthatarenowanessentialpartofmycareer.IamthankfultotheOf- ofScienceGraduateStudentResearch,USDepartmentofEnergy,andtheDissertation CompletionFellowship,MSUCollegeofNaturalScience,fortheircare. Mostimportantly,Icannotthankenoughmyparents,sister,andmybelovedYeni; withoutthem,nothingisasworthyasitis.Thanks,Father,foryourunconditionalguidance andcountlesslessonstoaimforhappinessandvalueinlife.Thanks,Mother,foryour exampleandpurelove.AndYeni,withoutyou,thisworkwouldhavebeenimpossibleand pointless.Everythingisforallofyou. IwanttothankallthefriendsIhadthepleasureofmeetingduringgraduateschool.And tomylife-longColombianfriends,thanksformakingthisjourneymorefun. Andlastly,thankyou,Universe,forenrichingandentertainingourliveswithsomany beautifulandexcitingproblemstobehopefullyunderstood. vi ThisworkwassupportedbytheUSDepartmentofEnergyunderContractNo.DE- FG02-08ER41546andContractNo.DE-SC0018636.Thisdocumentwaspreparedusingthe FermiNationalAcceleratorLaboratory(Fermilab)resources,aUSDepartmentofEnergy, ofScience,HEPUserFacility.FermilabismanagedbyFermiResearchAlliance,LLC (FRA),actingunderContractNo.DE-AC02-07CH11359. ThisresearchusedtheNationalEnergyResearchScienComputingCenter(NERSC) resources,aUSDepartmentofEnergyofScienceUserFacilityoperatedunderContract No.DE-AC02-05CH11231. vii TABLEOFCONTENTS LISTOFTABLES .................................... x LISTOFFIGURES ................................... xii Chapter1Introduction ............................... 1 1.1TheAnomalousMagneticMoment .......................2 1.1.1Muon g -2fromtheStandardModel ...................4 1.1.2Muon g -2:Theoryversusexperiment ..................10 1.2TheMuon g -2ExperimentatFermilab:ExperimentalMethod ........12 1.2.1Theanomalousmuonprecessionfrequency ! a .............13 1.2.2Muon g -2detectionsystems .......................18 1.2.3Themuon g -2magneticld .......................21 1.3BeamDynamics:Requirements .........................24 1.3.1Polarizationandbeamproductionperformance ............24 1.3.2Beamcharacterizationalongtheentirestoragering ..........26 1.3.3Momentumspreadandverticalmotion .................28 Chapter2TheBeamDeliverySystem(BDS) ................. 33 2.1Introduction ....................................33 2.2TheE989BeamDeliverySystem(BDS) ....................34 2.2.1Thepionproductiontargetstation ...................34 2.2.2M2/M3beamlines .............................36 2.2.3TheDeliveryRing(DR) .........................39 2.2.4M4/M5beamlines .............................41 2.3RealisticModelingoftheBDS ..........................42 2.3.1Beamlineelements ............................44 2.3.2Muonproductionfrompiondecay ....................45 2.3.3Nonlinearonbeamperformance .................48 2.3.4Spin-orbitcorrelations ..........................53 2.4Conclusions ....................................56 Chapter3BeamDynamicsattheMuon g -2StorageRing ......... 58 3.1Introduction ....................................58 3.2TheCOSY-basedMuonStorageRingModel ..................62 3.2.1Theelectrostaticquadrupolesystem ..................64 3.2.2Magneticdata ............................69 3.2.3Injectionkickermagnets .........................73 3.2.4Beamcollimation .............................78 3.2.5Initialbeamdistribution .........................79 3.3LinearBeamDynamics ..............................82 3.3.1Betatrontunes ..............................88 viii 3.3.2Opticallatticefunctions .........................92 3.3.3Closedorbits ...............................99 3.4NonlinearBeamDynamics ............................103 3.4.1Momentum-andamplitude-dependenttuneshifts ...........104 3.4.2Betatronresonances ...........................109 3.4.3Lostmuonmechanisms ..........................121 3.5NominalCharacteristicsoftheStoredBeam ..................127 3.5.1Beamboundaries .............................131 3.5.2Temporalmotionofthebeam ......................133 3.5.3Beamazimuthalmodulation .......................138 3.5.4Time-momentumcorrelations ......................142 3.6BeamDynamicsofRun-1 ............................144 3.6.1ReconstructionoftheelectricguideduringRun-1 ........147 3.6.2OpticallatticefunctionsduringRun-1 .................155 3.6.3Initialbeamdistributions ........................158 3.6.4ThesimulatedRun-1beam .......................164 3.7Conclusions ....................................169 Chapter4BeamDynamicsCorrectionStudies ................ 170 4.1Introduction ....................................170 4.2TheMuonLossCorrection ............................173 4.2.1Momentum-phasecorrelations ......................173 4.2.2Momentumspreadsubjecttolostmuons ................174 4.2.3Thecorrection C ml fromtheRun-1simulatedbeam ..........177 4.3ThePhaseAcceptanceCorrection ........................180 4.3.1Extractionofphasedrifts ........................182 4.3.2Phasedriftdrivingmechanisms .....................187 4.3.3Azimuthalextrapolationofphasedrifts .................191 4.4TheElectricFieldCorrectionandthePitchCorrection ............197 4.4.1Methodology ...............................198 4.4.2Electriccorrection C e ........................200 4.4.3Pitchcorrection C p ............................203 4.4.4Electricandpitchcorrections C e + C p ...............205 4.4.5Run-1considerations ...........................206 4.5TheWeightedMagneticField ~ B .........................209 4.5.1Themuon-weighted ~ B fromthesimulatedbeam .........211 4.5.2Methodtocalculate ~ B fromexperimentaldata .............218 4.5.3Methodvalidation ............................221 4.5.4Sensitivityof ~ B toazimuthalbeamvariations .............224 4.6Conclusions ....................................228 Chapter5Conclusions ................................ 230 BIBLIOGRAPHY .................................... 234 ix LISTOFTABLES Table1.1:Recent g -2valuesofchargedleptons,takenfrom[ 17 ]and[ 18 ].Tensions between g -2valuesfromexperimentsandtheoryhavebeenthemainmo- tivationof g -2experiments. ..........................4 Table1.2:Magnetparametersofthe g -2storagering. .................23 Table2.1:Second-ordertransfermapoftheDeliveryRing.Theecolumns correspondto( x j ,( a j ,( y j ,( b j ,and( l j ,whereasthelastcolumnindicates ( j 1 ;j 2 ;:::;j 7 )inthatorder.Forinstance,intheninthrow,column, the( x j xa )cotisdisplayed. ......................44 Table2.2:Numberofprotons( p ),muons( ),andpions( ˇ )alongtheBDS(quantities perprotonontarget). .............................49 Table2.3:RelativeinpionpopulationattheendoftheM3beamlinefrom fringeuptofourth-ordernonlinearterms,andrandommis- alignmentsofbeamlineelements(horizontalandverticalmisplacements withstandarddeviationsof0.25mm).Statisticalerrorsinthelastroware calculatedbasedonnumericalsimulationswithtentmisplacement scenarios,initializedwithindependentrandomseeds. ............53 Table2.4:RelativeenceinmuonpopulationattheendoftheM3,DeliveryRing (afterfourturns),andM5beamlinesectionsfromfringeup tofourth-ordernonlinearterms,andrandommisalignmentsofbeamline elements(horizontalandverticalmisplacementswithstandarddeviations of0.25mm).Statisticalerrorsinthelastrowarecalculatedbasedon numericalsimulationswithtentmisplacementscenarios,initialized withindependentrandomseeds. .......................53 Table3.1: a k; 0 cotsatthemainESQregion(seeEq.( 3.3 ))forHV= 18 : 3kV,scaledby r k + l 0 =k ! l !( r 0 =5cm).DuetotheESQmidplanesym- metry,cotswithodd l valuesarezero. ................68 Table3.2:Relativecomparisonsofcomputedhorizontaltuneswithmeasurements. .89 Table3.3:TunecocientsatHV=18 : 3kVuptoorder,normalizedby100 order . Forexample,( x j )= 0 : 13199935. .....................109 Table3.4:Resonantconditionsrelatedtobetatronexcitations. ............118 x Table3.5:StorageringnominaladmittancesforRun-1ESQsetpoints. ........132 Table3.6:MaximumvaluerangesinthestorageringalongtheazimuthforRun-1 settings. .....................................132 Table3.7:Transversemotionfrequenciesofthe g -2storedbeam(HV=18 : 3kV).The lastcolumnindicatescyclotronrevolutionsperfrequencycycle. ......134 Table4.1:TheRun-1beamdynamicscorrections[ 45 ].The C ml and C pa corrections areanticipatedtobetlysmallerforrunsposteriortoRun-1where theESQareexpectedtobehavestablyduringdatataking.TheE- correction C e counteractsthelargestbiasingin ! a duetoadditional spinprecessionsofstoredmuonsawayfrommagicmomentum. ......172 Table4.2: ! a correctionsduetomuonlossesfrombeamtrackingsimulations.Er- rorsforeachcalculationareproducedafterconsideringthemeasuredspin- momentumcorrelationuncertaintiesandsimulationstatisticallimits. ...180 Table4.3:Mainmechanismsofdetected-phasedrifts.Theverticalcontributiondomi- natesthephaseacceptancecorrectionduringRun-1atcalorimeters k =13 and k =19neartrackerslongitudinalacceptance. ..............188 Table4.4:Casesofstudyfor h ! a i tracking,fromthesimple(top)tothedetailed model(bottom). ................................203 Table4.5: C e versustracking(noverticalbetatronmotion). ..............203 Table4.6: C e + C p versustracking(fullbetatronmotion). ...............205 Table4.7:Deviationstotime-independent C e correctionsduringRun-1,basedon trackingsimulationsandopticallatticecalculationswiththeCOSY-based modelofthe g -2storagering. .........................209 Table4.8:Sensitivitystudiesofthemuon-weightedtoazimuthalbeamvariations. 226 xi LISTOFFIGURES Figure1.1:QEDvertexperturbativeloopsrepresentedwithFeynmandiagrams.The interactiondescribedbyDirac'sequationisrepresentedinthe0th-order loopdiagramshownin(a).In(b),theloopcontributes = 2 ˇ to a [ 15 ].Thediagramin(c)representsalltheotherhigherordercon- tributionsto a ofvirtualparticlescouplingtoleptonsorphotons. ...5 Figure1.2:Lowestorder(a)and(b)largestEWcontributionsto a SM . .......7 Figure1.3:FeynmandiagramsoftheLOhadronicvacuumpolarization(HVP)and light-by-lightscattering(HLbL). ......................8 Figure1.4:rencebetweenthetheoretical(green)andmostrecentmeasurements of a [ 28 ].Theinnertickmarksindicatethestatisticalcontributionto thetotaluncertainties.Thetheoreticalvaluefollowstherecommendation fromtheMuon g -2TheoryInitiative[ 22 ].The 4smallerexperimental errorgoalatFermilabrequirestwentytimesmorestatisticsandsystematic uncertaintiesreducedbyafactorof ' 3. ..................11 Figure1.5:Numberofpositronsfromahighlypolarizedmuonbeam( P =0 : 95) decayingvia + ! e + e ,subjecttothepositronenergy E inthelab- oratoryframeandignoringtheexponentialterm e ˝ .Theenergydis- tributionoscillatesasthemuonbeampolarizationrotatesrelativetothe beam'sdirectionofmotionwithanangularfrequency ! a independentof time.Theblackcurvecorrespondstothepolarizationparalleltomomen- tum,whereastheblueandredcurvesshowtheenergydistributionwhen thesetwodirectionsareperpendicularandanti-paralleltomomentum, respectively.Arrowsonthetop-rightcornerillustratepolarizationdirec- tionsrelativetomomentumdirection(gray),withthesamecolorscheme asthecurves.Theoscillationisstatisticallymaximizedwhenintegrating above ˘ 1 : 8GeV. ...............................16 Figure1.6:NumberofpositronsaboveenergythresholddetectedatFermilab'smuon g -2experiment,Run-1. ............................17 Figure1.7:(a)Strawtrackingstationand(b)adetectormodule[ 38 ]. ........20 xii Figure1.8:Muon g -2storageringhousedatFermilab.Thesteelyokesandsupercon- ductivecoilscrossedthousandsofmilesfromBrookhavenNationalLabo- ratorytoFermilabforextendeduse.Afterthedata-takingperiodat Fermilab,thestorageringmagnetwascoveredwiththermalinsulationto reducetemperaturegradientsthatecttheuniformityofthemagnetic ......................................21 Figure1.9:Crosssectionofthestorageringmagnet.Theyokeismadeofsixlayers ofmagnetsteelwithanopensidetoallowforpositronsfrommuondecay toreachtheinnersideofthering.Atotaloftwelveofthesesections assembledtogetherprovideacontinuousandultra-uniformmagnetic withinthepole'sgap,wherethestoredmuonbeamrevolvesforseveral thousandsofturns.Passiveandactiveshimmingviasurfacecurrents, thousandsofpiecesandaroundthemuonstorageregion,andothermov- ablepiecesofthestorageringmagnetallowtocalibrateandfurtherreduce inhomogeneitiesofthemagneticinthestorageregion[ 40 ]. .....22 Figure1.10:ArrayofpNMRprobesinstalledinthetrolley(whitecircles),usedto periodicallymeasurethemagneticinthestorageregion.Fixedprobes installedaboveandbelowthestorageregion(showninFig. 1.9 )areused tointerpolatethemagneticdatabetweentrolleyruns.Theheat maprepresentsatypicalsampleofthemagneticfromtrolleydata withamultipoleexpansion,withinthestorageregionandaveraged azimuthallyalongthering[ 39 ].ThecolorlegendunitsareTesla. ....24 Figure1.11:Horizontal(left)andlongitudinal(right)acceptancesofthetwo g -2straw trackerdetectorstations[ 43 ].Eachtrackerstationreadsdecayvertexes alongazimuthalsegmentsof ˇ 0 : 4mrad,whosemaximumacceptance peaksarepositionedatabout1mupstreamofthemodules. .......27 Figure1.12:Illustrationsofthepitch(left)and(right)corrections.Thepitch correctionemergesmainlyfromthetransverseverticalmotionofmuons inthelabframe;thestandardpitchcorrectionbecomesnon-negligibleat E989forthetypicalverticaloscillationsmaintainedbythestoragering (i.e., . 2mradverticaltiltanglesrelativetothehorizontalmidplane).In thecorrection,thespinofmuonsawayfromthemagicmomentum p 0 = m c= p a experiencesrotationsmostlycontainedinthetransversal planeasexpressedinthelasttermofEq.( 1.30 ).Themomentumaccep- tance ˘ 0 : 5%andtheelectric E y ˘ 6kV/cmat4cmfromtheideal orbit,providedbytheElectrostaticQuadrupolestations(ESQ)contribute toanoverallverticalprecessionfrequencyofabout400ppboppositeto thenominal g -2frequency. ..........................29 Figure1.13:SampleofthemuonspreaddistributionfromFastRotationanalysis[ 45 ]. 30 xiii Figure1.14:Verticalbeamfromthetrackingdetectionsystem[ 45 ]. ......31 Figure1.15:Beamtransverseintensityfromrealisticstorageringsimulations,referred asthefast-rotationsignal.Duetomomentumspread,therangeofcy- clotronfrequenciescauseshigh-andlow-momentummuonstorecombine asthebeamdecoheresinthelongitudinaldirection.Thecyclotronfre- quenciesdistributionisrecoveredfromaFouriertransform,whichyield themomentumspreaddistribution(seesampleinFig. 1.13 )forthe correction. ...................................32 Figure2.1:Aschematiclayoutofthebeamdeliverysystem(BDS).Secondaryparti- cles(mostlypions,muons,andprotons)downstreamofthetargetstation atAP0arecanalizedthroughtheM2/M3linesandinjectedintotheDe- liveryRing(DR),whereprotonsarediscardedandmostoftheremaining pionsdecayafterfourturns.Acleanerbeamofmostlymuonsisextracted totheM4/M5linesandultimatelydeliveredtotheMuon g -2StorageRing (SR). ......................................35 Figure2.2:Timestructureofthe10 12 -,120ns-pulsetrainimpingingtheproduction targetpercycle[ 55 ]. .............................35 Figure2.3:M2/M3betaanddispersionfunctions.Thehorizontaldispersion(solid greenline)originatedbythepulsedmagnetatthestartoftheM2line iscanceledoutbyaswitchdipole50mdownstream,whoselargepole widthaccommodatesthebeamtoentertheM3line.Ataround170m, the100mFODOchannel(i.e.,analternating-gradientarrangementof magneticquadrupolesfocusing(defocusing)anddefocusing(focusing)in thehorizontal(vertical)direction)isinterruptedbytwohorizontalbends (18 : 3 )toaligntheopticalaxiswiththedownstreambeamlineanda quadrupoletriplettocancelouthorizontaldispersion.Ontheright-hand sideoftheplot,theopticalfunctionsthegeometricandoptical matchingsectiontotransportthebeamtotheDeliveryRing,elevated approximately4ftabovefromtheM3line. .................37 Figure2.4:Numberofmuons, + ,pions, ˇ + ,andprotons, p + ,perprotonontarget (POT)alongtheM2/M3beamlines.Thehorizontalaxisrepresentsthe longitudinaldistanceoftheopticalaxis.Mainmuonlossesofabout11% and20%takeplaceatthe18 : 5 horizontalbend( s ˘ 160 : 0m)andalong theverticalinjectionupstreamoftheDR( s ˘ 280 : 0m),respectively.The star-shapedmarkeraround s ˘ 235mdepictsthenumberoftotalparticles perPOTfrommeasurements(Fig.3in[ 57 ]). ...............38 xiv Figure2.5:Numberofmuons, + ,pions, ˇ + ,andprotons, p + ,perprotonontarget (POT)alongfourturnsaroundtheDeliveryRing.Thedesignmomentum acceptanceof2%revealsnaturallyinbeamtrackingsimulationswhen accountingforthebeamlineaperturespresentintheDR.Allpionsdecay priortothethirdturnalongtheDR. ....................40 Figure2.6:Betaanddispersionfunctionsatonethree-foldsymmetricsectionofthe deliveryring(DR)wherethelongitudinaldistanceofthebeamlineis showninthehorizontalaxis.Fringemodeledinsimulationschange thebetafunctions, x and y ,bylessthan3%. ..............40 Figure2.7:M4/M5latticefunctionscalculatedwithCOSYINFINITY.Thedesignof theM4/M5beamlinesfavorslossesminimization. .............41 Figure2.8:Numberofmuons, + ,perprotonontarget(POT)alongtheM4/M5 beamlines.Thepale-rosedashedlinedepictsmuonswithinthemuon g - 2storageringmomentumacceptance.For g -2runsposteriortoRun-1, theinsertionofpassivewedgeabsorbersintheBDSmaximizedthemuon populationwithin j j < 0 : 5%toincreasethestoredbeamfraction[ 62 ]. .42 Figure2.9:MomentumspreadofthemuonbeamafterfourturnsaroundtheDelivery Ring.ThemomentumacceptancesustainedattheBDSbeamlinesallows forthepreparationofahighlypolarizedmuonbeam(P=0.969fromBDS simulations). .................................47 Figure2.10:Histogramsofthenormalizedmuonbeamspincomponents(innatural units)inopticalcoordinatesattheendofM3(left),DeliveryRing(center), andM5(right)beamlinesections.Thepolarizationprojections P x , P y , and P z (paralleltothe x , y ,and s directions,respectively)areshown insidethethickblackframes.Asindicatedbytheprojection P y =0,the highpolarization P ofthebeamislargelycontainedinthehorizontalplane. 49 Figure2.11:Simulationresultsofthenumberofpionsandmuonswith j j < 2%per POTunderuptofourth-order( o 4)alongtheM2/M3lineswith fringe(FR)turnedon.ThelongitudinaldistancealongtheM2/M3 linesisshowninthehorizontalaxis.Asdepictedbytheoverlappingcurves ofthemuonpopulationforthetwo o 1/ o 4cases,linearsimulationsdonot tlyfromresultswhenhigh-ordertermsinthecomputation oftheparticledynamicsaresimulated. ...................50 Figure2.12:EngefunctionofatypicaldipoleelementinCOSYINFINITY,used torepresentfringeeldsintheBDSsimulations.Theapertureforthis caseisequalto10cm. ............................51 xv Figure2.13:Fringeldonthepopulationofpionsandmuonsalongthede- liveryring(DR).Thehorizontalaxisrepresentsthelongitudinaldistance correspondingtofourconsecutiveturnsintheDR.FromtheCOSY-based BDSmodelsimulations,fringe(FR)contributetomaintainingmore pionswithintheaperturesoftheDR,whichconsequentlyincreasethe populationofmuonsby9 : 4%beforebeingextractedtotheM4line.The \o4"abbreviationinthelegenddenotestheorderofthecomputation(i.e., fourthorder). .................................52 Figure2.14:Simulationresultsofthemuonpopulationwith j j < 2%underthe ofbeamlinemisalignmentsalongtheM4/M5lines.Thenumberofmuons perprotonontarget(POT)isshownintheverticalaxisasafunction ofthelongitudinaldistancealongtheM4/M5lines.Theredlinedepicts thenumberofmuonsfortheidealcaseofnomisalignmentspresentin thebeamdeliverysystem.Thegreenbandsummarizessimulationresults forseveralscenariosofbeamlineelementsrandomlymisalignedinboth verticalandhorizontaldirections. ......................54 Figure2.15:Histogramofthemuonspinprojectionangleinthehorizontalplane withrespecttothereferenceopticalaxisatthestoragering(SR)en- trancefromsimulations.Thebeamdeliverysystem(BDS)isdesigned tofavorthecaptureoflongitudinallypolarizedmuonsfrompiondecay. However,asthemuonbeamtravelsthroughthebendingsectionsofthe BDS|especiallyalongtheDeliveryRingwhichhousesmultiplerectangu- larbendingmagnets|thepolarizationdevelopsatransversalcomponent beforethebeamisdeliveredtothe g -2storagering. ............55 Figure2.16:Beamaverageofthespinprojectionangleinthehorizontalplanewith respecttothereferenceopticalaxis, h ' a i ,versustheLorentzfactor, ,at theexitofthedeliveryringafterfourturns.Simulationresultspresented inthiscorrespondtoourCOSY-basedBDSmodelwithfringe turnedon. ...................................56 Figure3.1:LayoutoftheMuon g -2storageringatFermilab.Thefourlocationsofthe ESQarefoundnexttolabels\Q1-Q4"(eachcovering39 azimuthally) andlabels\K1-K3"indicatetheplaceoftheinjectionkickerplateswithin thering.\C"labelsdenotethebeamcollimatorsarrangement. ......60 xvi Figure3.2:FlowchartoftheCOSY-basedMuonStorageRingModel.Experimental measurementsconstitutetheguideasdinthemodel,whose mainoutputisnonlineartransfermapsforthecalculationofopticallat- ticefunctionsandbeamorbital(andspin)symplectictracking.With specialmethodsexplainedinthischapter,guidearereconstructed andinitialbeamdistributionsarerecreatedforadetailedcharacterization ofthemuonbeaminthe g -2storagering.Thecharacterizationisfurther usedforquanofbeamdynamicssystematiccorrectionsinthe g -2measurement. ............................63 Figure3.3:PhotographofoneESQstation.Forverticaltof(positive) muons,thetopandbottomplatesarepositivelychargedwhereasthelat- eralplatesarechargedwithnegativevoltage.Theverticalmagnetic inthestorageringlargelycontributestostablemotioninthehorizontal direction,inspiteofthedefocusingradialgradientfromtheESQinner andouterplates. ...............................64 Figure3.4:IllustrationoftheHVtracesfromplatesconnectedtothe1-pulse(red line)and2-pulse(bluecurve)HVsupplies.Themis-poweredESQplates (blue)duringtheinitial ˘ 7 safterinjectionshiftthebeamtowardsthe limitingaperturesofthestoragevolume,inordertoscrapethebeamin preparationforthedatatakingperiodat t> 30 s. ............65 Figure3.5:Radialelectricalongthemidplane(HV=18 : 3kV).Neartheaperture limitsat x =45mm,nonlinearitiesfromtheESQelectricdistortthe otherwiselinearradial .........................66 Figure3.6:Mainelectrostaticpotential[ kV ]fromanESQstationontheleftside, overlaidwithelectricvectorrepresentedwitharrows.Ontheright, electrostaticpotentialfromhigherordertermsintheTaylorexpansion; thecurvatureoftheplatesandthepresenceofthe20polearemanifested nearthelimitingboundsofthestorageregion. ...............69 Figure3.7:LongitudinalfringeatanESQstationedge.Thectivebound- aryextendstheoccupancyoftheelectricinthehard-edgeviewby ˘ 0 : 436%. ...................................70 Figure3.8:MagneticmultipolesfromTrolleyRun3956.Allthemultipoleterms areexpressedrelativetotheidealverticalmagnetic B 0 thatsustains magicmuonsintheidealorbit.Theskewdipoleterm(alsocalledradial wasmeasuredwithaHallprobein2016[ 40 ]andaveragedoutto recreatetheexpectedofthesurfacecorrectioncoils(SCC). ....72 Figure3.9:Transverseviewofakickerstation.Theplategeometryisdesignedto maximizecurrenconversioninthestorageregion. ........74 xvii Figure3.10:Representativekickerstrengthpulseduringtherunofthemuon g -2 experiment,measuredwithamagnetometer. ................75 Figure3.11:CrosssectionofthemagneticimplementedintheCOSY-basedmodel fortracking.Theisstrongerneartheplateedgesanduniformaround thecenter,tomuonsradiallyoutward. ...............76 Figure3.12:Pictureofacollimatorinsertedaroundthedesignorbit. .........79 Figure3.13:Simulatedbeamdistributionattheexitoftheafterthe focustopassthebeamthroughtheholeinthebacklegiron.Thebeamis mostlytangentialtothedesignorbitandabout77mmradiallyoutward atinjection. ..................................80 Figure3.14:Representativelongitudinalofthemuonbeamasmeasuredbythe T0detector.Itslengthisdesiredtobecontainedwithinthemainpeak ofthekickerpulsetomaximizebeamstorage.Thesolidlineshowsthe typeofmulti-Gaussianinterpolationusedinthemodeltorecreatethe longitudinalofthesimulatedmuonbeam. .............81 Figure3.15:Radial(black)andvertical(blue)betatrontunesasafunctionofESQ voltage. ....................................91 Figure3.16:Radial(black)andvertical(blue)linearchromaticitiesasafunctionof ESQvoltage. .................................92 Figure3.17:Radial , ,and functionsat5 s(redcurves),20 s(greencurves), and1000 s(blackcurves).Ontheleft-sideplotsHV=18 : 3kV,whereas HV=20 : 4kVforplotsontherightside.GrayshadowsdepictESQsta- tionsalongtheazimuth,wheretheQ1Supstreamedgeisat =0.Orange linesindicatecollimatorlocations.Redcurvesaresubjecttotheof theESQscrapingationandthegreencurveshavealmostreached theequilibriumvalues. ...........................94 Figure3.18:Vertical , ,and functionsat5 s(redcurves),20 s(greencurves), and1000 s(blackcurves).Ontheleft-sideplotsHV=18 : 3kV,whereas HV=20 : 4kVforplotsontherightside.GrayshadowsdepictESQsta- tionsalongtheazimuth,wheretheQ1Supstreamedgeisat =0.Orange linesindicatecollimatorlocations.Redcurvesaresubjecttotheof theESQscrapingationandthegreencurveshavealmostreached theequilibriumvalues. ...........................95 xviii Figure3.19:Radialdispersionfunctionat5 s(redcurves),20 s(greencurves),and 1000 s(blackcurves).TheESQvoltageisequaltoHV=18 : 3kVon theleft-sideplot,whereasHV=20 : 4kVfortheplotontherightside. Inhomogeneitiesinthenormalquadrupoletermofthemagneticbreak thefour-foldsymmetry. ...........................96 Figure3.20:Verticaldispersionfunctionat5 s(redcurves),20 s(greencurves), and1000 s(blackcurves).Ontheleft-sideplotHV=18 : 3kV,whereas HV=20 : 4kVfortheplotontherightside.Asthemagneticis mostlyorientedvertically,verticaldispersionsarenegligible. .......96 Figure3.21:Betafunctiondistortionsfrommagneticeldinhomogeneitiesat5 s(red curves),20 s(greencurves),and1000 s(blackcurves)forHV=18 : 3kV. 98 Figure3.22:Dispersionfunctiondistortionsfrommagneticinhomogeneitiesat 5 s(redcurves),20 s(greencurves),and1000 s(blackcurves)for HV=18 : 3kV.Theplotontherightsideshowsthetotaldistortioninthe verticaldispersionfunction,whichisequaltozerointhenominalcase, whereasontheleftsidetherelativedistortionoftheradialdispersion functionisdisplayed. .............................98 Figure3.23:Radialclosedorbits( =0)at5 s(redcurves),20 s(greencurves), and1000 s(blackcurves).Ontheleft-sideplotHV=18 : 3kV,whereas HV=20 : 4kVfortheplotontherightside.Theintentionalstretchingof theorbitduringscrapingincreasestheprobabilityofoutermostmuonsto hitacollimatorand,inthisway,minimizemuonlossrates. ........102 Figure3.24:Verticalclosedorbits( =0)at5 s(redcurves),20 s(greencurves), and1000 s(blackcurves).Ontheleft-sideplotHV=18 : 3kV,whereas HV=20 : 4kVfortheplotontherightside.Theinducedskewdipole ESQcreatedfromtheHVimbalancebetweenthetop/bottomplates shiftstheverticalclosedorbitforbeamscraping. .............102 Figure3.25:Representativeclosedorbitsduringthedatasetsoftherun(Run-1)of theexperiment(60h(1a),HK(1b),9d(1c),andEG(1d)).Fluctuations intemperaturethedipoletermsofthemagneticwhichled totclosedorbitsperRun-1dataset. ................103 Figure3.26:Amplitude-dependenttuneshifts( =0)withinthestorageregionwith- outmagneticimperfections(HV=18 : 3kV).Thehorizontalaxiscor- respondstoradialbetatronamplitudesandtheverticalaxisrepresents verticalamplitudes. ..............................107 xix Figure3.27:Amplitude-dependenttuneshifts( =0)withinthestorageregionwith magneticimperfections(HV=18 : 3kV).Thehorizontalaxiscor- respondstoradialbetatronamplitudesandtheverticalaxisrepresents verticalamplitudes. ..............................107 Figure3.28:Momentum-dependenttuneshifts(nobetatronamplitudes)withandwith- outmagneticimperfections(HV=18 : 3kV).Nonlinearshiftstake placefor j j > 0 : 2%,insidethemomentumacceptance. ..........108 Figure3.29:Beamradialde-coherencewith(left)andwithout(right)tuneshiftsat thereadoutlocationofthestrawtrackingdetector(station12).Asa reference,datafromthetrackerisshowninred.Thetwoplotsontop displayradialbeamcentroids,whereasthebottomplotscorrespondto radialbeamwidths. .............................110 Figure3.30:Fractionofmuonlossesbetween121 186 safterbeaminjectionwiththe COSY-basedmodelforseveralESQMeasurements[ 94 ]| showninarbitraryunits|resultfromdetectionsofminimumionizingpar- ticlesthatdeposit ˘ 170MeVofenergyintwoadjacentcalorimeterswhile twocollimatorswereinsertedtothestorageregion. ............111 Figure3.31:Anillustrationofresonanttunelinesandoperatingpoints(i.e.,ESQnom- inalvoltages)ofthe g -2storagering.Thehigh-voltageappliedtotheESQ platestosetoperatingpointsareshownnexttoresonancespredictedby numericalstudieswiththeCOSY-basedringmodel.Measurementsoflost muonshaverevealedresonancepeaksat ˘ 13 : 1 ; 16 : 8 ; 18 : 6 ; and21 : 1kVas well(seeFig. 3.30 ).Lostmuonsmeasurementshavenotbeenperformed forhigherESQvoltages.Redmarkersshowtheoperatingpointsusedin Run-1. .....................................113 Figure3.32:WiththetuneshiftsreadilyavailablefromtheCOSY-model,momentum andnormalformradiiofstoredmuonsfromarealisticinitialdistribu- tionasdescribedinSec. 3.2.5 areusedforthecalculationofthetune footprint.TheESQsetpointis18 : 3kV.Duetononlinearamplitude- andmomentum-tuneshifts,aconsiderablefractionofthestoredmuonsis bytheresonancearoundintunespace,especiallythe3 y =1res- onantconditiondrivenbytheskewmagneticsextupoleterm.Red(blue) markersindicatehigh(low)densityofentriesinthetunefootprint. ....115 Figure3.33:Fractionofmuonlossesfromsimulationsduringthetimeinterval121 186 safterbeaminjection.ndetailsarelistedonpage 116 . Thebottomdepictslostmuonfractionswithareducedvertical rangetodiscernlosseswhenmagneticimperfectionsarenotac- countedfor. .................................117 xx Figure3.34:Ontheleft,magneticskewsextupolecocient a 2 asmeasuredbyNMR probes.Ontherightplot,Fourierdecompositionofthecont a 2 in azimuthalharmonics a 2 = P n N =0 a 2 ;N cos( N + ˚ N ).The N =1term, whichisthemaindriveroftheresonantcondition3 y =1,isdepictedin greencolor. ..................................119 Figure3.35:Lostmuonsfractionoveramuonlifetimeafterthescrapingstageiscom- pleted.Inred(triangularmarkers),allthemagneticmultipoletermsare turnedonduringthebeamtrackingsimulation,whereasinblue(dia- mondmarkers),theseareturnedInsubsequentsimulations,itwas foundthatonlythemagneticskewsextupoletermfrommeasured inhomogeneitieswasienttodrivethebetatron-resonancepeakat HV ˇ 18 : 6kV. ................................120 Figure3.36:Phasespacecoordinatesandmaximumexcursionofamuonnotlost(ref- erencecase). ..................................122 Figure3.37:Trajectoriesinphasespaceofthreetmuons(distinguishedby color)inthelinearregime. ..........................123 Figure3.38:Maximumexcursionsofthreelostmuonsinthelinearregime. ......124 Figure3.39:Tunesofthreelostmuonsinthelinearregime.Notetheproximitybetween theredandblackentries(theredmarkerpartiallycoversthemarkerin blackcolor). ..................................125 Figure3.40:Trajectoriesinphasespaceofthreetmuons(distinguishedby color)inthenonlinearregime. ........................126 Figure3.41:Maximumexcursionsofthreelostmuonsinthenonlinearregime. ....127 Figure3.42:Tunesofthreemuonsinthenonlinearregimebeforegettinglost. ....128 Figure3.43:Muonlossfractionswithnonlinearities(blackcurve)andwithoutthem (graycurve)fromsymplectictracking,COSY-based,beamsimulations. TheESQsetpointwas18.3kV,twocollimatorswereinserted,andboth casesstartwiththesamedata-basedinitialdistribution.Muonlossrates arehighlytedbynonlinearitiesoftheguide ..........129 Figure3.44:Stroboscopictrackingintheverticalphasespaceillustratingorbitbehav- iorwithtwoperiod-3xedpointstructurespresent[ 101 ],forESQvoltage at18.3kV.Trajectoriesinblueandgreencolorsareexamplesofmuons (withintheringadmittanceandmomentumacceptance)withhighlymod- ulatedverticalamplitudes.PicturebycourtesyofAdrianWeisskopf. ..129 xxi Figure3.45:Muonlossratesfromseveraltrackingsimulations(HV=18.3kV).The ofdamagedresistorsandthefromsymplecticenforcement duringtrackingisshown. ...........................130 Figure3.46:Qualitativecomparisonbetweenstoredmuonratesfromsimulations(gray) at t =186 safterbeaminjectionandmeasuredrelativepositronrates (arbitraryunits)fromstoredmuondecays[ 107 ].Errorbarscorrespond tostandarderrorsofthemultiplenumericalanalysesforeachESQvolt- ageAstheverticalandradialadmittancesincreaseand decrease,respectively,proportionaltotheESQvoltage,thefractionof storedmuonsincreases,reachingaplateauat22kVwherethestored fractionstartstobecomemoresensitivetotheradialadmittance.The lowfractionofstoredmuonsisaconsequenceofthemomentumspread beingabouttwotimeslargerthanthemomentumacceptance,thedisper- sionmismatchbetweentheendoftheM5-lineandthestoragering,and theimperfectkickspurposedtoinjectthebeam. ..............131 Figure3.47:Long-termradialbeamde-coherence. ....................134 Figure3.48:Long-termverticalbeamtemporalmodulations. ..............135 Figure3.49:BeamfrequenciesextractedwithFFTfromcentroidsmotion. ......136 Figure3.50:BeamfrequenciesextractedwithFFTfromwidthsmotion. ........136 Figure3.51:Comparisonbetweenclosedorbitandbeamcentroidfromtrackingat157 downstreamfromtheentranceofQ1S.Trackingdataisrandomizedto removebeambeatingfrommismatch.Theinitialdistributionandguide arepreparedforthe60 h caseasexplainedinSec. 3.6 . .......137 Figure3.52:Comparisonbetweenbeamwidthfromopticallatticefunctions(greenand redentries)andbeamwidthfromtrackingat157 downstreamfromthe entranceofQ1S.Thearediscussedinthemaintext.Tracking dataisrandomizedtoremovebeambeatingfrommismatch.Theinitial distributionandguidearepreparedfortheRun-1acaseasexplained inSec. 3.6 .Theredlinecorrespondstoacasewithconstantemittances, whichintheverticalcaseplaysaroleindescribingaccuratelyvertical beamwidths. .................................137 Figure3.53:Beamemittancesfromtracking.Duetothesmallerverticaladmittance, theverticalemittanceismorebymuonlossrates. ........138 xxii Figure3.54:Radialphasespaceatlatetimesofthedatatakingperiod(left)andits spatialprojection(right)fromRun-1simulation.Thepatternclosely resemblesobservationsanditscharacteristicskewnessispresentinall Run-1datasets. ................................139 Figure3.55:Verticalphasespaceatlatetimesofthedatatakingperiod(left)andits spatialprojection(right)fromRun-1simulation. .............140 Figure3.56:Comparisonbetweenclosedorbits(redcurves)andrandomizedbeamcen- troidsfromRun-1trackingsimulations. ...................141 Figure3.57:Comparisonbetweenbeamwidthsfromopticallatticefunctions(redcurves) andrandomizedbeamwidthsfromRun-1trackingsimulations. .....141 Figure3.58:TypicalRun-1momentumtimebeamdistributionfromRun-1tracking simulation. ..................................143 Figure3.59:ProjectionsofthetypicalRun-1momentumtimebeamdistributionfrom Run-1trackingsimulation. ..........................143 Figure3.60:RadialCBOfrequenciesduringthefourRun-1datasetsfromstrawtrack- ingdetectorsdata.Semi-transparentmarkersareobtainedfromsliding sinusoidal( 5 s)totherecordedradialbeamcentroid,whereassolid linesresultfrommulti-parameterthroughtheentireBothmethods areequivalent. ................................145 Figure3.61:VerticalcentroiddriftsduringRun-1fromstrawtrackingdetectorsdata. Solidlinesarewithdoubleexponentialtermsandaconstantpartas thefunctionalform. .............................146 Figure3.62:Single-CADDOCKhigh-voltageresistor(top)andchainofpottedHV resistors(bottom).Twoofthelattertypeofresistorsbecamedamaged duringRun-1. .................................147 Figure3.63:HV-tracessample(circlemarkers)fromHVprobemeasurementsinSeptem- ber2018,atQ1Lplatesconnectedtothedamagedresistors.Blueandred linesdepictnominalHVtraces. .......................148 Figure3.64:HighvoltageonQ1LTandQ1LBplates(left)andobjectivefunction (right)periterationastheoptimizationtakesplacefortheRun-1,EndGame dataset,at40 s.TheoptimizersupportedbyCOSYINFINITY(i.e.,the generalizedleastsquaresNewtonmethod)Q1LT/Q1LBHVssuchthat theCOSY-basedstorageringmodelwithdamagedresistorsaccountedfor reproducesaverticalpoint(equivalenttotheverticalmean)and CBOfrequencyasmeasuredbytrackerstation12. ............152 xxiii Figure3.65:ReconstructedHVtraces(normalizedtotheirnominalvoltageset-points) atQ1LduringRun-1datasets.DashedandsolidlinescorrespondtoQ1LT andQ1LBplates,respectively. ........................152 Figure3.66:Comparisonsbetweentrackerdata(bluemarkers)andtrackingsimula- tions(redmarkers)attrackerS12azimuthalreadoutlocationwiththe damaged-resistorseimplemented.OntheleftistheCBOfrequency, whereasontherightverticalcentroidsareshown.Similarly,strongagree- mentsareobtainedfortheotherRun-1datasets.Forcomparisonpur- poses,theverticalentriesontherightplotareshiftedwithintrackermis- alignmenterrors. ...............................153 Figure3.67:Blackmarkersareverticalbeamdrifts(40-300 s)fromcalorimeterdata (energythresholdof1.7GeV)undercorrections,where doubleexponentialwereemployed.Inblue,verticalclosedorbitdis- tortiondriftsfrom40-300 susingtheCOSY-basedmodelwiththere- constructedHVtracesimplemented.Theblueerrorbandisanestimate oftheuncertaintyintroducedbytheverticaldispersion.Theredsquares arefroma gm2ringsim muontrackingsimulation[ 45 ]withanimple- mentationoftheelectricguidereconstructedwiththeCOSY-based model.Thesimulationsarematchedtodatabyassociatingtheclosed orbitplaced ˘ 22 upstreamofthecalorimeterposition. .........154 Figure3.68:Verticalclosedorbitsat30 s(redcurves),100 s(bluecurves),200 s (greencurves),and300 s(blackcurves)duringRun-1.Grayshadows depictESQstationsalongtheazimuth,wheretheQ1Supstreamedge isat =0.Orangelinesindicatecollimatorlocations.Redcurvesare subjecttotheoftheESQscrapingandthegreen curveshavealmostreachedtheequilibriumvalues. ............156 Figure3.69:Verticalbetafunctionsat30 s(redcurves),100 s(bluecurves),200 s (greencurves),and300 s(blackcurves)duringRun-1. .........157 Figure3.70:Radialbetafunctionsat30 s(redcurves),100 s(bluecurves),200 s (greencurves),and300 s(blackcurves)duringRun-1. .........158 Figure3.71:Radialdispersionfunctionsat30 s(redcurves),100 s(bluecurves), 200 s(greencurves),and300 s(blackcurves)duringRun-1. .....159 Figure3.72:Relativeverticalbetafunctionsdriftfrom30 sto1000 s. ........160 Figure3.73: x , y , D x ,andverticalclosedorbitat210 fromtheupstreamentrance ofstationQ1SfordatasetRun-1d. .....................161 xxiv Figure3.74:Beamdriftsfrom40 sto300 sanchoredtotrackerdatafortheRun- 1ddataset.GrayshadowsdepictESQstationsalongtheazimuth,where theQ1Supstreamedgeisat =0.Orangelinesindicatecollimator locations.WiththeRun-1opticallatticefunctions,theobservedbeam driftsmeasuredbythemuon g -2strawtrackingdetectorsareprojected alongtheentireazimuthofthestoragering. ................162 Figure3.75:Resolution-andacceptance-correctedradialpositionversustimefrom trackerdata,Station12.Byreconstructingthetrajectoryofdecaypositrons detectedbythetrackingplanes,muondecaypositionsareextrapolated. Coherentoscillationsareaconsequenceofthebeaminjectionprocess. ..163 Figure3.76:Reconstructedbeamdistributionintheradialdirection.Ontheleftplot, theupperlimitsaredeterminedbythephysicalaperturesofthecolli- matorswhichboundmaximumradialexcursions,whereasthemaximum kickerstrengththelowerlimits. ...................164 Figure3.77:Muonbeamintensityfromsimulation(left)andtrackerdata(right),sta- tion12,duringRun-1afrom30 s 30 s. isinjectedintothestorageringforabout7 s,atwhichpointthesecondpulsetakesplace tochargethemuptothenominalHVvalue.Thetransitionsfrommis-poweredtonominal HVsareadiabatic,wheretheresistorsconnectedtotheplatesarechosentoyieldtransition timesof ˝ C = RC ˇ 5 s.Thetotalto-groundcapacitanceoftheplates,ontheorderof 100 300pF[ 74 ],arenoteasilymalleablesincetheysurgefromseveralpropertiesonthe 65 grounding,intervenedbyHVleads,feed-throughs,platesgeometry,etc.Since ˝ C ˛ T C (where T C ˇ 0 : 1492 sisthecyclotronperiod),beamtrackingsimulationsarereliably performedwithone-turntransfermapspreparedpercyclotronperiod.Aftercommissioning, thetypicalHVsetpointsofthe g -2runs(HV ˘ 18kV)werechosensothattheESQ canstablyrunwithoutdischargingsparksand,ontheotherhand,totrytoavoidbetatron resonances.Theresultingfocusing/defocusingelectricreachvaluesontheorderof 6kV = cm. DuetothelongitudinalcurvatureandoverallgeometryoftheESQelectrodes,thegener- atedelectricexhibitnonlinearfeatureswhoseisperceivedmorenoticeablynear thelimitsofthestoragevolume,asshowninFig. 3.5 .Toreproducearealisticrepresenta- Figure3.5:Radialelectricalongthemidplane(HV=18 : 3kV).Neartheaperturelimits at x =45mm,nonlinearitiesfromtheESQelectricdistorttheotherwiselinearradial tionoftheESQelectricthetransverseelectrostaticpotentialisasatransverse Taylorexpansionintheradial-vertical(or,equivalently, x - y )planeatalongitudinallocation s : V = V ( x;y;s )= 1 X k =0 1 X l =0 a k;l ( s ) x k y l k ! l ! : (3.2) 66 Byexploitingthehorizontal-planesymmetryoftheESQstation'scrosssection,allthe a k;l ( s ) cotsareuniquelydeterminedoutofthemidplanecots a k; 0 ( s )asfollows[ 60 ]: a k;l +2 = a 00 k;l kha 00 k 1 ;l + kh 0 a 0 k 1 ;l a k +2 ;l (3 k +1) ha k +1 ;l 3 kha k 1 ;l +2 k (3 k 1) h 2 a k;l 3 k ( k 1) h 2 a k 2 ;l +2 k ( k 1) 2 h 3 a k 1 ;l k ( k 1)( k 2) h 3 a k 3 ;l +2 ; (3.3) where h istheinverseofthestationcurvatureradius(7 : 112m)and a 0 k;l = da k;l =ds .The resultingsetofcotsLaplace'sequationincurvilinearcoordinatestoensure aFullyMaxwellianThe a k; 0 cotsatthemainESQregionarecalculated withconformalmappingmethods[ 71 ]andspinCOSYINFINITY,whichcomputes thefroma3Dout-of-planeexpansionviaDAmethods[ 75 ]forthecomputationof transfermaps. Table 3.1 showsthe a k;l termssptoanominalESQstationchargedatHV=18 : 3kV. Thefour-foldsymmetricgeometryoftheplatesaroundthestationcenterfavorsthepresence oftheleadingquadrupole(k=2),12pole(k=6),20pole(k=10),andfurther4(2 i 1)pole midplane-symmetric(l=0)terms,beingthequadrupoleterm|whichlargelythebe- tatrontunesofthering|about3 : 4%higherthanthevoltageappliedtotheplates[ 76 ].Of specialinterestisthe20pole,whoselargerrelativemagnitudemanifestsitselfinbetatron resonanceandtuneshifts,aselaboratedinthenextchapter.Figure 3.6 showstheoveralland nonlinearstructuresoftheelectrostaticpotentialfromachargedESQstationimplemented intheCOSY-basedmodel. Theeboundary( z EFB =1 : 2195cm)andfringeoftheESQarecal- culatedusingCOULOMB's[ 50 , 71 ]boundary-elementmethodeldsolver.Figure 3.7 shows 67 Table3.1: a k; 0 cotsatthemainESQregion(seeEq.( 3.3 ))forHV=18 : 3kV, scaledby r k + l 0 =k ! l !( r 0 =5cm).DuetotheESQmidplanesymmetry,cotswithodd l valuesarezero. k l =0 l =2 l =4 l =6 l =8 l =10 0 01.900E+04-9.388E 01-4.796E+012.489E 02-7.298E+02 1 0-4.006E+024.621E 029.103E+00-3.997E 033.848E+02 2 -1.900E+045.633E+007.194E+02-6.969E 013.284E+04-6.926E+01 3 0-5.281E 02-4.046E+013.240E 02-5.540E+036.986E+00 4 0-7.194E+021.280E+00-1.533E+054.594E+025.058E+04 5 01.517E+01-2.861E 021.357E+04-2.512E+01-9.601E+03 6 4.796E+01-2.133E 011.533E+05-6.485E+02-1.517E+059.198E+02 7 02.199E 03-7.387E+032.167E+011.829E+04-5.919E+01 8 0-3.284E+042.061E+021.517E+05-1.165E+03-3.715E+06 9 06.923E+02-4.174E+00-1.174E+045.152E+015.659E+05 10 7.298E+02-9.739E+00-5.058E+044.990E+023.715E+06-4.505E+04 11 01.014E 012.327E+03-1.510E+01-3.989E+052.474E+03 12 04.598E+03-6.226E+01-1.576E+062.296E+04-1.050E+02 13 0-9.696E+011.216E+001.151E+05-9.276E+023.665E+00 14 -5.053E+011.363E+002.598E+05-4.643E+032.941E+01-1.096E 01 15 0-1.426E 02-1.169E+041.341E+02-7.778E 012.885E 03 16 0-1.299E+043.063E+02-3.088E+001.784E 02-6.826E 05 17 02.740E+02-5.871E+006.028E 02-3.649E 041.475E 06 18 8.491E+01-3.853E+009.206E 02-1.036E 036.788E 06-2.950E 08 theresultingfringewhichisimplementedintheCOSY-modelESQtransfermaps viaanEnge-function: F ( z )= 1 1+exp a 1 + a 2 ( z=D )+ ::: + a 6 ( z=D ) 5 ; (3.4) where D isthefullapertureoftheESQ( D =10cm)andtheEngecotsare a 1 =0 : 14389529 ;a 2 =6 : 85939851 ;a 3 = 1 : 87096936 ; a 4 =0 : 80158053 ;a 5 = 0 : 40704326 ;a 6 =0 : 06588881 : (3.5) Withthefringetheleadingelectricmultipolecotstrengthsarescaled accordinglyandthelongitudinaldependenciesarecapturedrecursivelyinCOSYINFINITY fortheESQmodeling.Theveboundaryforthisspcaseelongatesthe extensionofeachstation,reducingradialtunesby ˘ 0 : 07%andincreasingtheverticalones 68 Figure3.6:Mainelectrostaticpotential[ kV ]fromanESQstationontheleftside,overlaid withelectricvectorrepresentedwitharrows.Ontheright,electrostaticpotentialfrom higherordertermsintheTaylorexpansion;thecurvatureoftheplatesandthepresenceof the20polearemanifestednearthelimitingboundsofthestorageregion. by ˘ 0 : 05%withrespecttoahard-edgecase. Twoofthe32HVresistorsconnectedtotheESQplatesweredamagedduringthe data-takingrun(\Run-1")ofthemuon g -2experiment.Consequently,theresultingelectric exhibitedspecialbehaviorsthatthetransversestabilityofthestoredbeam, asdescribedinSec. 3.6 . ForthetransfermapspreparationintheCOSY-basedmodel,theODEssubjecttothe ESQelectricdescribedabovealsoaccountfortheembeddingmagneticinthe storagering.TheimplementationandfeaturesofthemagneticldsintheCOSY-based modelaredescribednext. 3.2.2Magneticdata Thesubtleimperfectionsofthemagneticwithinthestorageregionoftheringdrive closedorbitdistortionsandbetatronresonancesatsphigh-voltagesettingsoftheESQ. 69 Figure3.7:LongitudinalfringeatanESQstationedge.Theeboundary extendstheoccupancyoftheelectricinthehard-edgeviewby ˘ 0 : 436%. Inourmodel,suchinhomogeneitiesareincludedbasedonofnuclearmagneticresonance (NMR)trolleyrunmeasurementsfromthe g -2FieldTeam.Theazimuth-independent providetheextractionof2Dmagneticnormalandskewmultipolecots(uptothe decapoleterm)byattributingthescalarmeasurementstothepredominantverticalmagnetic ByconsideringLaplace'sequationincylindricalcoordinatesandomittingazimuthal z variationsofthemagneticpotential V , r V ( r;;z )= 1 r @ @r r @V @r + 1 r @ @ 1 r @V @ =0 ; (3.6) theverticalmagneticisexpressedasasummationofmagneticmultipolecots: B y ( r; )= v 0 + 1 X n =1 r r 0 n [ v n cos + w n sin ] : (3.7) InEqs.( 3.6 )and( 3.7 ), r = p x 2 + y 2 isthetransversedistancefromthedesignorbitand 70 r 0 isthereferenceradiusofthemultipoles.Theangle isthecounterclockwiseanglefrom the x axissuchthatcos = x=r . SincetheNMRmeasurementsofthescalarmagneticareinsensitivetoitsdirection and,ontheotherhand,theisknown apriori tobedominantlyvertical,asensible approximationoftaking B y asthescalaritselfyields B ( r; ) ˇ B y ( r; )= B 0 b 0 + X n =1 r r 0 n [ b n cos( )+ a n sin( )] ! : (3.8) Duetotheppm-levelradialandlongitudinalcomponentsofthemagneticinthestorage ring[ 40 ],thisapproachaddsasmallerrorofonly ˘ 10ppbtoNMRmeasurements[ 77 ].In thedeterminationoftheld,themainerrorsoriginatefromtransients,temperature motionals,calibrationandpositioninguncertaintiesoftheprobesthat mapthewithinthestoragevolume(referto[ 39 ]).However,thankstothehighunifor- mityofthetosomeextent,sucherrorsweresmallforRun-1,i.e., ˘O (100ppb)relative tothemaindipoleandareexpectedtobereducedforRun-2andposteriorruns. TherelativestrengthsofthemagneticmultiplecomponentsimplementedintheCOSY- basedmodelareshowninFig. 3.8 andfollowthenotationin[ 78 ]: B y + i B x = B 0 X n ( b n + ia n )( x + iy ) n : (3.9) Inthemodel,thetransfermapofanESQstationofazimuthallength L withmagnetic inhomogeneitieswithin( M E;B )ispreparedasasequenceofadjacentopticalelements 71 Figure3.8:MagneticmultipolesfromTrolleyRun3956.Allthemultipoletermsare expressedrelativetotheidealverticalmagnetic B 0 thatsustainsmagicmuonsinthe idealorbit.Theskewdipoleterm(alsocalledradialwasmeasuredwithaHallprobe in2016[ 40 ]andaveragedouttorecreatetheexpectedofthesurfacecorrectioncoils (SCC). asfollows: M E;B ( L )= M E ( L i = 2) M L ( l ) M B ( ˚;l ) M E ( L i = 2)+ C x ( L i ;˚ )+ C y ( L i ;˚ ) n : (3.10) Themap M E ( L i )encapsulatestheFullyMaxwellianESQelectric(seeSec. 3.2.1 )em- beddedintheidealverticalmagnetic B 0 = p =eˆ 0 where p 0 isthemagicmomentum, e theelementarycharge,andthedesignradiusis ˆ 0 ˇ 7 : 112m.Theazimuthallength L i (typ- icallyabout0 : 1 )ischosensothatthemaplengthcorrespondsto L andtheazimuthal variationofthemagneticmultipolesiscaptured.Byvirtueofthethinlensapproximation| andthentlysmallsizesofthemagneticinhomogeneitiesrelativetothemain verticalanelement M B ( ˚;l )withasuperpositionoftheazimuthallydependentnor- 72 malandskewmagneticmultipolesattheazimuth ˚ actsonthemaptocoverthe ofthemagneticerrors;thelength l isequalto1 mandthemultipolesareproperly scaled.Amulti-Gaussianfunctionisimplementedtocalculatethemagnitudesofthemea- suredmagneticmultipolesatanyazimuth.Adriftelement M L ( l )ofnegativelengthis forcounteractingthenonzerolengthof M B ( ˚;l ).Thelow-orderskewandnormal magneticcots, a 0 and b 0 ,aremodeledassymplectickicksandactontheconstant part(writtenexplicitlyas C x and C y inEq.( 3.10 ))ofthetransfermapsothat a = B 0 b 0 L i =˜ m ; b = B 0 a 0 L i =˜ m ; (3.11) where ˜ m = B 0 ˆ 0 isthemagneticrigidity. a and b aretheradialandverticaldirection inparticleopticalcoordinates.TopreservetheHamiltonian(viaphase-spacepreser- vation)duringbeamtrackingsimulationswiththisdetailedimplementationofthesubtle magneticerrors,thesymplecticconditionisenforcedinCOSYINFINITY. 3.2.3Injectionkickermagnets Threekickerstations(seeFig. 3.9 )1 : 27mlongandabout1 = 4ofradialbetatronwavelength downstreamoftheexit,wherethebeamentersthering,areinstalledinthestorage ring(seeFig. 3.1 )forbeaminjection.Themuonbeamemerges ˘ 77mmtangentiallyshifted fromtheidealorbit.Inorderfortheinjectedmuonstoendupwithintheringstorage volume,thekickerstationsarerequiredtosteerthemabout10 : 5mradradiallyoutward. Tothisend,thekickerplatessupplyanintegratedmagneticof1 : 1kG : moppositely directedtotheverticalofthestoragering.Thekickerstationsneedtopulseat100Hz synchronizedtothe120nspulsetrainfromthebeamlines(seeFig. 2.2 ),andshutbefore 73 Figure3.9:Transverseviewofakickerstation.Theplategeometryisdesignedtomaximize currenconversioninthestorageregion. thepulserevolvesthe149ns-longstoragering[ 79 ].Inviewofthesetimingspa robustpulserandlow-impedancekickerstationsaredesiredtomeettherequiredriseand decaytimes. Duetoimpedancemismatch,thetemporalshapeofthekickerstrengthsexhibitsaringing structureasthesignalistedafterthemaintransientisinduced.Figure 3.10 shows atypicalsampleduringtherunofthemuon g -2experiment[ 80 ]. IntheCOSY-basedmodel,eachinjectedmuonisateachlongitudinalcenterof thethreekickerstationsperturn.Thesteeringisappliedassymplectickicksinbothradial andverticaldirections: a ( x;y;t ) ˇ e L p 0 B k y ( x;y;t ) ; b ( x;y;t ) ˇ e L p 0 B k x ( x;y;t ) ; (3.12) where L isthedesignkickerlength.Thetransversestructureofthetransient ~ B k is 74 Figure3.10:Representativekickerstrengthpulseduringtherunofthemuon g -2ex- periment,measuredwithamagnetometer. basedonavailable OPERA simulationsofthe x y kickersgeometryenclosedbythestorage ringvacuumchamber[ 81 ];Fig. 3.11 showsthevectorThetimedependenceof ~ B k is capturedbasedonmagnetometermeasurementsofthekickerstrength(seeFig. 3.10 ),scaled accordingtothespkickersettingsofthe g -2dataperiodbeinganalyzed;typicalsummed kickerstrengthsduringtheproductionrunswerearound130kV.Furtherimprovements canbemadeontheimplementationofEqs.( 3.12 ),namelymultiplescaledkicksalongeach kickerstationandtheadditionofmomendependence. Forspintracking,a3 3rotationmatrix R ( ~z i )(duetoelectric ~ E 0 andmagnetic ~ B 0 toactontheinitial ~s i =( s x ;s y ;s z )spincoordinatesofmuonsinopticalcoordinates ~z i =( x i ;a i ;y i ;b i ;l i ; )(seeSec. 3.3 foraofthecoordinates)isobtainedfromthe 75 Figure3.11:CrosssectionofthemagneticimplementedintheCOSY-basedmodelfor tracking.Theisstrongerneartheplateedgesanduniformaroundthecenter,to muonsradiallyoutward. T-BMTequation[ 47 ] ~! 0 S = e m 1 + a ~ B 0 a 1+ ~ ~ B 0 ~ a + 1 1+ 1 c ~ ~ E 0 ˙ (3.13) where ~! 0 S isthespinprecessionfrequencyinthelaboratoryframe.The ~z variable isthe particlemomentumrelativeto p 0 and l isproportionaltothetimeoftrelativeto themaximumpeakofthebeam'stimewhichcoincidesinturnwiththemaximum kickerstrengthatstationK2inCOSY-basedtrackinginjectionstudies.Thetimingbetween thesetwoisnoteasilydetermined,althoughnobetweenthemisagoodproxy forstoragefractionoptimization. IntheFrenet-Serretframerelativetothereferenceparticle,theaxisofrotationvector ~u ofthespin-rotationmatrixisfromEq.( 3.13 ).Furtherexpressingtheinstantaneous 76 precessionangle 0 intermsofatialarc-lengthadvance,thematrixisas R ( x;a;y;b; )= 0 B B B B B @ cos 0 + u 2 x 1 cos 0 u x u y 1 cos 0 u z sin 0 u x u z 1 cos 0 + u y sin 0 u y u x 1 cos 0 + u z sin 0 cos 0 + u 2 y 1 cos 0 u y u z 1 cos 0 u x sin 0 u z u x 1 cos 0 u y sin 0 u z u y 1 cos 0 + u x sin 0 cos 0 + u 2 z 1 cos 0 1 C C C C C A (3.14) where u x = ! 0 Sx ! 0 S ;u y = ! 0 Sy ! 0 S ;u z = ! 0 Sz ! 0 S (3.15) and dt = dL v = 1+ x ˆ 0 ds 0 v ) 0 = ! 0 S dt = ! 0 S 1+ x ˆ 0 ds 0 v : (3.16) Thenotation ds 0 referstothenominalarc-lengthadvanceinatimeinterval dt ; ˆ 0 isthe radius, v theparticlespeed,and x theradialrelativetothedesignorbit.In termsofopticalparticlecoordinates,momenta p i and i = v i =c canbeintermsof ~z i coordinates: p x = x = p 0 a = ) x = p 0 mc a = 0 0 a p y = y = p 0 b = ) y = p 0 mc b = 0 0 b p z = z = p 0 q (1+ ) 2 a 2 b 2 = ) z = 0 0 q (1+ ) 2 a 2 b 2 ; (3.17) whichyield ! 0 Sx = 0 v 0 B 0 ˆ 0 1 + a B 0 x a 1+ ~ ~ B 0 x a + 1 1+ 1 c h ~ ~ E 0 i x ˙ = 0 v 0 B 0 ˆ 0 ( 1 + a B 0 x a ( 0 0 ) 2 (1+ ) aB 0 x + bB 0 y a + E 0 y c 0 0 a + 1 1+ q (1+ ) 2 a 2 b 2 ) ; (3.18) 77 ! 0 Sy = 0 v 0 B 0 ˆ 0 1 + a B 0 y a 1+ ~ ~ B 0 y a + 1 1+ 1 c h ~ ~ E 0 i y ˙ = 0 v 0 B 0 ˆ 0 ( 1 + a B 0 y a ( 0 0 ) 2 (1+ ) aB 0 x + bB 0 y b E 0 x c 0 0 a + 1 1+ q (1+ ) 2 a 2 b 2 ˙ ; (3.19) and ! 0 Sz = 0 v 0 B 0 ˆ 0 ˆ a 1+ ~ ~ B 0 z a + 1 1+ 1 c h ~ ~ E 0 i z ˙ = 0 v 0 B 0 ˆ 0 ( a ( 0 0 ) 2 (1+ ) aB 0 x + bB 0 y q (1+ ) 2 a 2 b 2 0 0 a + 1 1+ a E 0 y c b E 0 x c !) ; (3.20) where ~ E 0 = E 0 x ^ x + E 0 y ^ y and ~ B 0 = B 0 x ^ x + B 0 y ^ y .Fortheinjectionkickerscase, ~ B 0 = ~ B k and ~ E 0 =0. 3.2.4Beamcollimation Thecomponentsthattheboundsofthestoragevolumeintheringaretheso-called collimators;seeFig. 3.12 .Theseringsaremadeofcopper,withinnerandouterradiiof 45mmand50mmrespectively.Duetotheirlowmagneticsusceptibilityof 9 : 63 10 6 , theonthesurroundingmagneticeldwhentheyareinsertedorretractedisequally negligible[ 82 ]. Duringthe ˘ 50turnsafterbeaminjection,themainpurposeofthecollimatorsis toscrapethemuonswiththelargesttransverseexcursionswhiletheapplicationofasym- metricvoltagesalongtheESQstationsdistortstheclosedorbits.Furthermore,thecircular geometryofthecollimatorscontributestominimizingmuonlosseswhiledataistakenby 78 Figure3.12:Pictureofacollimatorinsertedaroundthedesignorbit. tlyexcludingthenonlinearelectricfromtheESQbeyondthecollimatoraper- tures.Fortheformerintent,muonshittingcollimatorsloserigiditybytransferringenergy tothecollimators,madeofcopper.Moreover,oncethesemuonsstartdivergingfromthe storagevolumetowardstheinnersideofthering,theydosoquicklyenoughsothattheir correspondingemittedpositronsdonotenterintothesignalfromwhich ! a isextracted. Thereareecollimatorstationsalongthestoragering,asshowninFig. 3.1 .During therunofFermilab'smuon g -2experiment,onlytwocollimatorswereinsertedaround theidealorbittocompensatefortheinitiallylower-than-nominalkickersinjectionscheme. Afterward,allthe5collimatorswereinserted. UsingtoolsinCOSYINFINITY,themuonbeamistreatedasanarrayofvectorswhich permitstotlycollimatemuonsbeyondwaperturesduringtracking. 3.2.5Initialbeamdistribution AsexplainedinChap. 2 ,theacceleratorcomplexatFermilabdelivershighlypolarizedmuon bunchestothe g -2storageringat100Hz.Ahandfulofnumericalstudieswitht 79 programsrecreatedthebeamtransportedalongtheMuonCampusduringthecommissioning ofthemuon g -2experiment[ 52 , 56 , 83 ],whosecoordinatedallowedtobenchmarkeach package.Forbetatronresonancescansandmomentum-timecorrelationstudies,anexternal distribution[ 84 ]wastakenforsubsequentbeamanalysiswiththeCOSY-basedstoragering model.Thisdistributionresultsfromtransferring G4beamline numericalsimulations[ 56 ] attheendoftheM5linethroughfringetotheentranceoftheringbacklegiron[ 85 ]. Themapsofthesearesuperimposedwiththethroughthematerial-freeyoke volumewhereasuperconductingisplacedtocancelouttheaccountingforthe parameters(e.g.,angleandd)thatmaximizethestoredfraction.Atthispoint, themismatchedbeamiscentered ˘ 77mmradiallyoutwardfromtheidealorbit.Figure 3.13 illustratesthebeamdistributionsinphasespaceandmomentum-timespread.Theinitial Figure3.13:Simulatedbeamdistributionattheexitoftheafterthefocusto passthebeamthroughtheholeinthebacklegiron.Thebeamismostlytangentialtothe designorbitandabout77mmradiallyoutwardatinjection. timeofthebeamissetbasedonmeasurementsfromscintillatingdetectors[ 35 ].Two photomultipliertubes(PMTs)ontheverticalsidesofascintillatorreadouttheresulting light,whichisadjustedwithdensitytoensureanoperationalrange.Priortoentering thestoragering,theincomingbeamcrossesthe\T0"detector,whichmeasuresitsintensity andlongitudinalForbeamtrackingsimulations,aninterpolationtoT0data|from 80 theaverageofarepresentativetrainofeightpulsesduringtherunsoftheexperiment| replacedtheoriginaltimefromthe G4beamline distributiontransferredthrough theor(seeFig. 3.14 ).ThemaximumpeakoftheT0signalisalignedwiththerays Figure3.14:RepresentativelongitudinalproofthemuonbeamasmeasuredbytheT0 detector.Itslengthisdesiredtobecontainedwithinthemainpeakofthekickerpulseto maximizebeamstorage.Thesolidlineshowsthetypeofmulti-Gaussianinterpolationused inthemodeltorecreatethelongitudinalofthesimulatedmuonbeam. originallywithnoht,anditsrelativetimingwiththecentralkickermaximum strengthissettozero;thisproxyrelationagreeswiththeexperimentaltuningtomaximize muonstorageviathisparameter.Thetimecoupledwiththeinjectionkickerringing signalsdeterminesthecorrelationbetweenhtandmomentumofthestoredmuon beam. 81 3.3LinearBeamDynamics Bytreatingthe g -2storageringasanopticalsystem,thebeamdynamicsframeworklaysout practicalconceptualtoolsforathoroughcharacterizationofthestoredbeam.Toachieve afullextentofsuchcharacterizationforfurther a systematicerroranalysistotheppb level,theactionofnonlinearguideonthemuonbeamneedstobeaccountedfor(see Sec. 3.4 ).Nevertheless,theconsiderationofelectricandmagneticinthestoragering uptoorder(i.e.,dipoleandquadrupolecomponentsinaTaylorexpansiondescriptionof theintermsoftheopticalparticlecoordinates)encompassesmostofthethat thebeamorbitalmotionalongthering.Eventhoughtheinjectionprocessfor themuonstogetintothestorageringyieldsthestoragevolume(i.e.,amostlycirculartoroid withsectionalradius r 0 ˇ 45mmandoverallradius ˆ 0 ˇ 7 : 112mm)almostfullofmuons, thesmalladmittancelimitedbytheinsertedcollimatorsandaratherweakfocusingallows tosatisfytheparaxialapproximationand,thisway,thefollowinglinearmatrixformalism. Inthisregime,theequationsofmotionarelinearizedsothattheopticalcoordinates ofanindividualmuon ~z ( 2 )= f x 2 ;a 2 ;y 2 ;b 2 ;l 2 ; g arebyitsinitialstate ~z ( 1 )= f x 1 ;a 1 ;y 1 ;b 1 ;l 1 ; g ,where 1 and 2 aretwoazimuthallocationsinthering: z i ( 2 )= 6 X j =1 z i j z j z j ( 1 )+ C i : (3.21) InEq.( 3.21 ), z i isthe i thcoordinateofthevector ~z where x and y arethehorizontal andverticalspatialdeviationsfromthereferenceorbit 1 ; a = p x =p 0 and b = p y =p 0 are themomentumdeviationsin x and y relativetothereferencemomentum p 0 (inthiscase 1 Referenceorbit(alsoknownasidealorbit)referstothecirculartrajectorywithradius ˆ 0 ,fullyhorizontal, alignedwiththestorageringcenter,andintheverticalcenteroftheESQstations. 82 themagicmomentum m c= p a ˇ 3 : 094GeV = c); l = ( t t 0 ) v 0 0 = (1+ 0 )aparameter proportionaltothet t t 0 relativetoareferenceparticlewith t = t 0 ,speed v 0 andLorentzfactor 0 ;and =( p p 0 ) =p 0 themomentumdeviation.The z i j z j and C i termsarecoetsderivedfromtheequationsofmotionanditiscustomarytoarrange themalgebraicallyinalinearmatrix M (1 ! 2)suchthat ~z ( 2 )= M (1 ! 2) ~z ( 1 ) ; (3.22) orinextendedform: 0 B B B B B B B B B B B B B B B B B @ x a y b l 1 C C C C C C C C C C C C C C C C C A 2 = 0 B B B B B B B B B B B B B B B B B @ ( x j x )( x j a )( x j y )( x j b )( x j l )( x j ) ( a j x )( a j a )( a j y )( a j b )( a j l )( a j ) ( y j x )( y j a )( y j y )( y j b )( y j l )( y j ) ( b j x )( b j a )( b j y )( b j b )( b j l )( b j ) ( l j x )( l j a )( l j y )( l j b )( l j l )( l j ) ( j x )( j a )( j y )( j b )( j l )( j ) 1 C C C C C C C C C C C C C C C C C A 0 B B B B B B B B B B B B B B B B B @ x a y b l 1 C C C C C C C C C C C C C C C C C A 1 + 0 B B B B B B B B B B B B B B B B B @ C x C a C y C b C l C 1 C C C C C C C C C C C C C C C C C A : (3.23) Thisformalismpermitstotreatthe g -2storageringasanopticallatticemadeofasequence ofopticalelements,whereeachoftheseelementsisrepresentedwithacorrespondingmatrix bytheguidealongitsazimuthalspan.Inthestoragering,therearefourtypes ofopticalelementsintheabsenceofmagneticerrorsorESQplatemisalignments(see Fig. 3.1 forvisualization): \ DIEQS ":ShortESQstation( DIEQS =13 +2 EFB long). \ DIS ":ShortmagneticsectionbetweenashortESQstation(upstream)andalong ESQstation(downstream)( DIS =4 2 EFB long). 83 \ DIEQL ":LongESQstation( DIEQL =26 +2 EFB long). \ DIL ":LongmagneticsectionbetweenalongESQstation(upstream)andashort ESQstation(downstream)( DIL =47 2 EFB long). AsexplainedinSec. 3.2.1 ,theangle EFB = z EFB =ˆ 0 accountsforthee boundaryattheedgesofESQstations.Inthisbutrepresentativeapproach, DIL and DIS arehomogeneoussectormagnetsdescribed[ 60 ]by M DI ( )= 0 B B B B B B B B B B B B B B B B B @ cos ˆ 0 sin 000 ˆ 0 (1 cos ) sin =ˆ 0 cos 000sin 001 ˆ 0 00 000100 0 0 +1 sin 0 0 +1 ˆ 0 (1 cos )001 ˆ 0 h 0 1 0 0 0 +1 sin i 000001 1 C C C C C C C C C C C C C C C C C A ; (3.24) where istheazimuthallengthofthesectorand ˆ 0 =7 : 112mthenominalbendingradius. Inasimilarmanner,the DIEQ elementscorrespondtoinhomogeneoussectormagnets[ 50 ] wheretheinhomogeneityindex n e |whichintroducesalineardependencetothee bending B y ( x )|emergesfromtheESQelectricquadrupolegradient: B y ( x )= B 0 1 n e x ˆ 0 (3.25) = B 0 1 ˆ 0 v 0 B 0 @E x @x x ˆ 0 : (3.26) B 0 = p 0 =eˆ 0 ˇ 1 : 4513Tistheverticalnominalandtheradialelectricgradient 84 @E x =@x horizontallydefocusesthepositivelychargedmuonbeam.Thesolutionofthe linearizedequationsofmotionforsuchopticalelementproducethefollowingresult: M DIEQ ( )= 0 B B B B B B B B B B B B B B B B B @ cos( p 1 n e ) ˆ 0 p 1 n e sin( p 1 n e )000( x j ) p 1 n e ˆ 0 sin( p 1 n e )cos( p 1 n e )000( a j ) 00cos( p n e ) ˆ 0 p n e sin( p n e )00 00 p n e ˆ 0 sin( p n e )cos( p n e )00 ( l j x )( l j a )001( l j ) 000001 1 C C C C C C C C C C C C C C C C C A ; (3.27) where ( x j )= ˆ 0 1 n e [1 cos( p 1 n e )] ; (3.28) ( a j )= 1 p 1 n e sin( p 1 n e ) ; (3.29) ( l j x )= 0 0 +1 1 p 1 n e sin( p 1 n e ) ; (3.30) ( l j a )= 0 0 +1 ˆ 0 1 n e [1 cos( p 1 n e )] ; and(3.31) ( l j )= ˆ 0 0 0 +1 (" 1 1 n e 1 (1+ 0 ) 2 # 1 (1 n e ) 3 = 2 sin( p 1 n e ) ) : (3.32) Withthesetransfermatrices,thecoordinatesvector ~z ofamuoncanbetransferredthrough thering.Forinstance,thecoordinates ~z i ofamuonupstreamofashortESQstationafter n fullrevolutionsevolveaccordingtotherepeatedactionofeachringelementasfollows: ~z n = M 0 n ~z i = M DI ( DIL ) M DIEQ ( DIEQL ) M DI ( DIS ) M DIEQ ( DIEQS ) 4 n ~z i (3.33) AtHV=18 : 3kV(atypicalESQhighvoltageduringRun-1ofthemuon g -2experiment), 85 theone-turnmatrix M 0 1 is 2 M 0 1 = 0 B B B B B B B B B B B B B B B B B @ 0 : 92045 E +00 0 : 25743 E +010 : 00000 E +000 : 00000 E +000 : 00000 E +000 : 48687 E +00 0 : 45556 E 010 : 95902 E +000 : 00000 E +000 : 00000 E +000 : 00000 E +00 0 : 35324 E +00 0 : 00000 E +000 : 00000 E +00 0 : 60304 E +000 : 18795 E +020 : 00000 E +000 : 00000 E +00 0 : 00000 E +000 : 00000 E +00 0 : 41356 E 01 0 : 36936 E +000 : 00000 E +000 : 00000 E +00 0 : 34732 E +00 0 : 44242 E +000 : 00000 E +000 : 00000 E +000 : 10000 E +01 0 : 49502 E +02 0 : 00000 E +000 : 00000 E +000 : 00000 E +000 : 00000 E +000 : 00000 E +000 : 10000 E +01 1 C C C C C C C C C C C C C C C C C A : (3.34) Asshownin M 0 1 ,theradialandverticalmotionintheringislinearlydecoupled(e.g., ( x j y )=( x j b )=( y j x )=( y j a )=0andanyothermapcomponentsthatrelateradialand verticalcoordinates).Expectedly,themomentumdeviation isnotbytheother orbitalcoordinates(and C =0)beingthestorageringatime-independentHamiltonian system.Furthermore,onecandemonstratethatthetransversecoordinates x and y donot divergesince j ( x j x )+( a j a ) j < 2 ; j ( y j y )+( b j b ) j < 2 : (3.35) Themap M 0 1 doesnotaccountfortheppm-levelmagneticinhomogeneities.When theseareintroducedinthecalculationofthemapasexplainedinSec. 3.2.2 basedonNMR- probemeasurements,constantterms(i.e., C i termsinEq.( 3.23 ))emerge.Thenonzero termsin M 0 1 slightlychangebylessthan0 : 1%asaconsequenceofthenormalquadrupole termsfrommagneticeldimperfections.Andmoreover,magneticskewquadrupoleterms addweakcouplingbetweentheradialandverticalmotionvianonzero( x j y ),( x j b ),( a j y ), ( a j b ),etc.terms.Suchnewmap M 0 1 relativetotheidealorbitinthestorageringcanbe re-expandedaroundtheclosedorbitoriginatedbythemagneticdimperfections.Eachof 2 Inthefollowingdiscussion, -relatedcomponentsaresometimesexpressedintermsof k =( K K 0 ) =K 0 instead,where K isthekineticenergyforcompatibilitywithCOSYINFINITY'ssetofparticlecoordinates. Forthemomentumacceptanceandreferencevalueinthe g -2storagering, and k donotbymore than( 0 +1) 0 1 ˇ 3 : 41%. 86 the ~z 0 ( ; )= f x 0 ( ) ;a 0 ( ) ;y 0 ( ) ;b 0 ( ) ; 0 ; g ( )momentum-dependentpointsatany azimuth thatcomposethisnewclosedorbitcanbefoundwiththefollowingoperation: ~z 0 ( ; )=( M 0 ( ) I ) 1 ~ 0 ; (3.36) where I istheidentitymap, M 0 istheoriginalmaparoundthereferenceorbitat ,and ~ 0 containsthe ~z coordinatesofthereferenceorbit ~ 0= f 0 ; 0 ; 0 ; 0 ; 0 ; g . Inordertoextracttheindex,Twissparameters,anddispersionfunctions(See Sec. 3.3.2 ),aswellastoperformsymplecticbeamtrackingandnonlinearcalculations,maps suchas M 0 1 arere-expandedaroundtheircorrespondingpoints.Thisway,theresulting mapisorigin-preserving(i.e.,thevector ~ 0isunchangedundertheactionof M ).Forthis purpose,andtaking M 0 1 asanexample,themaparoundthepoint M 1 isfoundas follows: M 1 ( )= M 0 1 ( )( ~z 0 ( ; )+ I ) ~z 0 ( ; ) : (3.37) Inthelinearregime, M 1 = M 0 1 .However,underthepresenceofnonlinearterms,thatis notthecase;nonlineartermsleakinsidethelineartermsof M 1 when ~z 0 isnonzero(referto Sec. 3.4 forthediscussionofnonlinearbeamdynamicsinthemuon g -2experiment).The resultingone-turnmap M 1 forNMR-probedataduringthelastRun-1datasetis M 1 = 0 B B B B B B B B B B B B B B B B B @ 0 : 92089 E +00 0 : 25742 E +01 0 : 43513 E 03 0 : 23697 E 030 : 00000 E +000 : 48210 E +00 0 : 45431 E 010 : 95891 E +000 : 26600 E 040 : 73226 E 040 : 00000 E +00 0 : 35194 E +00 0 : 88037 E 030 : 65890 E 02 0 : 60221 E +000 : 18825 E +020 : 00000 E +000 : 90815 E 02 0 : 13050 E 04 0 : 12671 E 03 0 : 41352 E 01 0 : 36789 E +000 : 00000 E +000 : 26431 E 03 0 : 34600 E +00 0 : 44366 E +00 0 : 35668 E 03 0 : 83647 E 020 : 10000 E +01 0 : 49482 E +02 0 : 00000 E +000 : 00000 E +000 : 00000 E +000 : 00000 E +000 : 00000 E +000 : 10000 E +01 1 C C C C C C C C C C C C C C C C C A : (3.38) 87 Withtheselinearmaps,beamtrackingsimulationsareplausiblewiththelimitation ofoverlookingnonlinearsuchasbetatronoscillationsde-coherence(i.e.,nonlinear amplitude-andmomentum-dependenttuneshifts)andbetatronresonance(presented inthesubsectionsbelow).Moreover,theinformationcontainedinthe( z i j z j )termsallows forthecharacterizationofseveralbeamparameters,asshownnext. 3.3.1Betatrontunes Itiscustomaryinthemuon g -2collaborationtoparametrizesomedynamicalproper- tiesofthebeambyvisualizingthestorageringasasingleinhomogeneoussectormagnet M DIEQ (360 )(refertoEq.( 3.27 )).Inthisperspective,whichispossiblethankstothe ratherweakfocusinginthering,theindex n e isscaledbytheESQstationsoccupancy inthering'sazimuthtoobtain: n = 4( DIEQS + DIEQL ) 4( DIEQS + DIS + DIEQL + DIL ) n e =0 : 4376998 n e : (3.39) DuringthefourRun-1datasets,thetwonominalESQvoltageswereHV=18 : 3kVand HV=20 : 4kVwhichcorrespondtothefollowing g -2indices: n =0 : 108790931and0 : 121275137 : (3.40) 88 Forcomparisonpurposeswith M 0 1 (Eq.( 3.34 )),atHV=18 : 3kVtherepresentativelinear mapis M DIEQ (360 )= 0 B B B B B B B B B B B B B B B B B @ 0 : 93882 E +00 0 : 45717 E 010 : 00000 E +000 : 00000 E +000 : 00000 E +000 : 47213 E +00 0 : 45717 E 010 : 93882 E +000 : 00000 E +000 : 00000 E +000 : 00000 E +00 0 : 35279 E +00 0 : 00000 E +000 : 00000 E +00 0 : 48084 E +000 : 18906 E +020 : 00000 E +000 : 00000 E +00 0 : 00000 E +000 : 00000 E +00 0 : 40664 E 01 0 : 48084 E +000 : 00000 E +000 : 00000 E +00 0 : 35279 E +00 0 : 47213 E +000 : 00000 E +000 : 00000 E +000 : 10000 E +01 0 : 49560 E +02 0 : 00000 E +000 : 00000 E +000 : 00000 E +000 : 00000 E +000 : 00000 E +000 : 10000 E +01 1 C C C C C C C C C C C C C C C C C A : (3.41) Asetofbetatrontunes( x ; y astheaveragenumberoftransverseoscillations perturn|canbecalculatedfromthetracesof M DIEQ (360 ): x = 1 2 ˇ cos 1 ( x j x )+( a j a ) 2 = p 1 n (3.42) y = 1 2 ˇ cos 1 ( y j y )+( b j b ) 2 = p n; (3.43) whereforHV=18 : 3kV: x =0 : 944038701and y =0 : 329834703 : (3.44) InTable 3.2 ,relativecomparisonswithmeasurements[ 86 ]areshown(i.e.,( x data x ) data x ). Table3.2:Relativecomparisonsofcomputedhorizontaltuneswithmeasurements. HV[kV] M DIEQ M 0 1 M 1 18.3 -0.0557%-0.0108%-0.0053% 20.4 -0.0701%-0.0137%-0.0034% Radialtunesobtainedfromtheapproximatedmodel M DIEQ (360 ),fromtheidealring model( M 0 1 )andfromaringwithmagneticinhomogeneities( M 1 )bylessthan 89 0 : 08%relativetotunesextractedfromtrackerdataduringRun-1,being M 1 themostac- curatecase;accountingforthe 30ppmbackgroundofthenormalmagneticquadrupole componentyieldsthesuperiordescriptionofthetunes.AsexplainedinSec. 3.6 ,failures intheinstrumentationoftheESQledtotime-dependenttuneswhichevolvedbyabout 0 : 5%duringthemeasurementperiod.Thus,fortherelativeinTable 3.2 the stabletuneswereextractedfromtheobservedfrequencies f CBO ofthebeamradialbetatron oscillationatlatetimesofthewhere f CBO = f c (1 x )(3.45) andthecyclotronfrequency f c istakenas f c = p 0 2 ˇm 0 ˆ 0 =6 : 704958MHz : (3.46) Asexplainedin[ 50 ],fringeattheESQedgesdistortthetunesbylessthan0.0008%. OthersuchasESQplatemisalignmentswhicharenotconsideredinthepresented modelscouldbethereasonforthesmalldeviationfromexperimentaldata.Byvaryingthe voltageontheESQstations,thesetpointtocontrolthesetofbetatrontunesisadjusted. Figure 3.15 showsseveralcomputationsofthetunesforseveralESQvoltages.Itisdesired toavoidESQvoltagesthatsatisfyresonantconditions(seeEq.( 3.66 )),whichunderthe presenceofthehigh-ordercomponentsofthering'smainmagneticandESQelectric couldinduceunstabletransversemotionasexplainedinSec. 3.4 . Ofparticularinterestforbeamde-coherenceandbetatronresonanceanalysesarethe 90 Figure3.15:Radial(black)andvertical(blue)betatrontunesasafunctionofESQvoltage. linearchromaticities ˘ x;y ,as ˘ x;y x;y (3.47) where x;y aretuneshiftsduetoamomentum .Tothem,momentum- dependentpoints ~z 0 ( ; )arecomputedandthediagonaltermsofthemaprelative tothepointsareusedtocalculatethemomentum-dependenttunes: x ( )= 1 2 ˇ cos 1 ( x j x )( )+( a j a )( ) 2 = 0 x + ˘ x + ; (3.48) y ( )= 1 2 ˇ cos 1 ( y j y )( )+( b j b )( ) 2 = 0 y + ˘ y + : (3.49) Inthisway,linearchromaticitiesforseveralESQvoltagesarecalculated;seeFig. 3.16 for theresults. 91 Figure3.16:Radial(black)andvertical(blue)linearchromaticitiesasafunctionofESQ voltage. 3.3.2Opticallatticefunctions Withtheorigin-preservinglineartransfermapsinhand,periodicfunctionsthatthe beamstructureatequilibriumcanbe[ 58 ].Thisparametrizationresultsfromthe guidesinthestoragering,whichinmoregeneraltermsisanopticallatticethatacts onthemuonstreatedasrays.Therearethreeofsuchfunctionscommonlyused: x ( )= ( x j x )( ) ( a j a )( ) 2sin2 ˇ x ; x ( )= ( x j a )( ) sin2 ˇ x ; x ( )= ( a j x )( ) sin2 ˇ x : (3.50) Theangle indicatestheazimuthallocationinthering,andthenumeratorcomponents ontheright-handsideareextractedfromone-turnmapsat .Theverticalsituationis analogoustotheverticalcase.Withtheguidesimplementation,theone-turnmapsare preparedalongtheringazimuthandthenrotatedaroundtheirpointsfortheextraction oftheopticallatticefunctions. ( ), ( ),and ( )canalsobetransferredwiththeorigin- preservingmapcomponentsbetweentoazimuths,thoughtheformercasewaschosenfor 92 computational.Figures 3.17 and 3.18 showtheselatticefunctionsalongthe g -2 storageringforringduringRun-1. Duetothemomentumspreadofthestoredmuonbeam,thedispersionfunction D x;y ( ) isalsonecessarytodescribethebeam.Thefollowingperiodicconditionisbythe dispersionfunction: 0 B B B B B @ D D 0 1 1 C C C C C A = 0 B B B B B @ ( x j x )( x j a )( x j ) ( a j x )( a j a )( a j ) 001 1 C C C C C A 0 B B B B B @ D D 0 1 1 C C C C C A ; (3.51) whichimplies D x ( )= (1 ( a j a )( ))( x j )( )+( x j a )( )( a j )( ) 2 ( x j x )( ) ( a j a )( ) : (3.52) InFigs. 3.19 and 3.20 dispersionfunctionsforseveralstorageringsettingsarepresented. Thelongitudinalderivativeofthedispersionfunction, D 0 ,relatestotheofmomentum on a and b ,andisnotrelevantforthecharacterizationintendedinthisdissertation. Forthebeamextrapolationaroundtheringbasedonbeamdiagnosisattwoazimuthal locations,thefollowingrelationsbetweenbeamwidths(interpretedasRMSsandsymbolized with ˙ )andlatticefunctionsareused: ˙ 2 x ( ;t )= " x ( t ) x ( ;t )+ D 2 x ( ;t ) ˙ 2 ; (3.53) wherethetimedependenceofthewidth, anddispersionfunctionsemergefromevolving guidesuchasRun-1duringproductionperiod(seeSec. 3.6 ).Themomentumdistri- butionspread ˙ isobtainedfromFastRotationanalysis[ 87 ].Fromrealisticsimulations 93 Figure3.17:Radial , ,and functionsat5 s(redcurves),20 s(greencurves),and 1000 s(blackcurves).Ontheleft-sideplotsHV=18 : 3kV,whereasHV=20 : 4kVforplots ontherightside.GrayshadowsdepictESQstationsalongtheazimuth,wheretheQ1S upstreamedgeisat =0.Orangelinesindicatecollimatorlocations.Redcurvesare subjecttothectsoftheESQscrapingandthegreencurveshavealmost reachedtheequilibriumvalues. 94 Figure3.18:Vertical , ,and functionsat5 s(redcurves),20 s(greencurves),and 1000 s(blackcurves).Ontheleft-sideplotsHV=18 : 3kV,whereasHV=20 : 4kVforplots ontherightside.GrayshadowsdepictESQstationsalongtheazimuth,wheretheQ1S upstreamedgeisat =0.Orangelinesindicatecollimatorlocations.Redcurvesare subjecttothectsoftheESQscrapingandthegreencurveshavealmost reachedtheequilibriumvalues. 95 Figure3.19:Radialdispersionfunctionat5 s(redcurves),20 s(greencurves),and1000 s (blackcurves).TheESQvoltageisequaltoHV=18 : 3kVontheleft-sideplot,whereas HV=20 : 4kVfortheplotontherightside.Inhomogeneitiesinthenormalquadrupoleterm ofthemagneticbreakthefour-foldsymmetry. Figure3.20:Verticaldispersionfunctionat5 s(redcurves),20 s(greencurves),and 1000 s(blackcurves).Ontheleft-sideplotHV=18 : 3kV,whereasHV=20 : 4kVforthe plotontherightside.Asthemagneticismostlyorientedvertically,verticaldispersions arenegligible. 96 withtheCOSY-basedmodel,inworst-casescenariosaccountingformuonscraping, ˙ varies bylessthan0 : 8%duringthemuonthesensitivitylevelofradialwidthstothisscaleof momentumspreadvariationsissmallcomparedtotheoverallazimuthalwidthmodulations, i.e.,ontheorderof0 : 03mm.Fortheanalogousverticalcase,thedispersionfrommeasured magneticgradientsisnegligible(seeFig. 3.20 )forthebeamcharacterization.Thetransverse emittance " x istheRMSofthebeamdistributioninphasespace: " x = q h x 2 ih a 2 ih xa i 2 : (3.54) Withthetwostrawtrackingdetectorsdataofthebeamtransversecoordinatesovertime togetherwithEq.( 3.53 ),theemittanceisquanallowingtoprojectthebeamwidthsin thisway.Forverticalwidthprojections,trackerdataatonlyonelocationistsince D y ˇ 0. ThemainmodulationsofthelatticefunctionsareproducedbytheESQelectricand theverticalmagneticMagneticimperfectionsmostlyfromthenormalquadrupole componentcontributetoadditionalmodulationsintheopticalfunctions,ofabout0 : 5% orlessinrelationtothecaseofaperfectmagnetic(seeFigs. 3.21 and 3.22 ).Since theextractedmultipolecotsfromtrolleydata(exceptdipoleterms)changedbyless than100ppbrelativeto B 0 ,theextradistortionsfrommagneticimperfectionsarenot expectedtochangeonarun-by-runbasis.Ontheotherhand,ESQplatemisalignments canaddgradienterrorsaswell.Anexternalanalysisbasedonalignmentsurveydata[ 88 ] havedeterminedthemtobesimilarinsizeasdistortionsfrommagneticerrors.For Run-1analysis,theimplementationof(andectsfrom)magneticinhomogeneitiesare necessaryforthebeammeansandwidthscalculationaroundthering;thefromESQ 97 Figure3.21:Betafunctiondistortionsfrommagneticldinhomogeneitiesat5 s(red curves),20 s(greencurves),and1000 s(blackcurves)forHV=18 : 3kV. platemisalignmentsaretreatedaserrors. Figure3.22:Dispersionfunctiondistortionsfrommagneticinhomogeneitiesat5 s(red curves),20 s(greencurves),and1000 s(blackcurves)forHV=18 : 3kV.Theplotonthe rightsideshowsthetotaldistortionintheverticaldispersionfunction,whichisequaltozero inthenominalcase,whereasontheleftsidetherelativedistortionoftheradialdispersion functionisdisplayed. Theframeworkprovidedbythe anddispersionfunctionsasusedinEq.( 3.53 )is wellconsolidatedforbeamswithellipticaldistributionsinphasespaceandmatched|i.e., aligned|withtheinvariantellipseofthemachine(inthiscasethestoragering)aroundthe 98 point,nedby: x ( x D x ) 2 +2 x ( x D x )( a D 0 x )+ x ( a D 0 x ) 2 = A; (3.55) where A isaconstant.Duetothechallengingbeaminjection,thestoredbeamismismatched totheinvariantellipse.Asaresult,coherentbetatronoscillations(whichmostlyvanishat latetimesofthemeasurement-periodcycles)producetemporalbeatingaroundthelattice functions.Inspiteofthemismatching,thelatticefunctionsrecreatethebeammodulations aroundtheringfor ! a systematiccorrectionsandthemuonweighting,asexplainedin Sec. 3.5 .Theotheringredientsforafulldescriptionofthemuonbeamalongtheazimuth areclosedorbitdistortionspresentednext. 3.3.3Closedorbits Asdinthissectionandwithintheframeworkoftransfermaps,closedorbitsarethe setofpoints ~z 0 ( ; )for0 < 2 ˇ thatremainunchangedafterbeingtransferred aroundonefullrevolution( M 0 ( ) ~z 0 ( ; )= ~z 0 ( ; )).Theseareobtainedvia ~z 0 ( ; )= ( M 0 ( ) I ) 1 ~ 0for( M 0 ( ) I )invertible,asinthe g -2ringcase.Whenthemuoncoordinates coincidewiththoseofapoint,itfollowsthetrajectorybytheclosedorbitin theabsenceofexternalperturbations.Furthermore,forastablesystem(seeEq.( 3.35 )), muonswithinthedynamicapertureandapartfromtheclosedorbitoscillatearounditby virtueoftheguiderestoringforces.Inthelinearcase,thematchedmuonbeamcanbe 99 statisticallyaveragedtoacentroid z ( ; )anditsspatialcoordinateswillobey x ( ; )= x 0 ( ; ) ; y ( ; )= y 0 ( ; ) : (3.56) Duetothesmallmomentumacceptanceinthe g -2case,anexplicitdistinctionfromthe momentum-dependentxedpointcanbemadewithtprecision,byaccountingfor thelinearmomentumdependenceofthepoint: x ( )= x 0 ( )+ D x ( ) ; y ( )= y 0 ( ) : (3.57) Similartothebeamwidthscase,amismatchedbeamexhibitstransverseoscillationsofits centroidsaroundtheclosedorbit.Inthiscase,theindividualbetatron(i.e.,transverse) amplitudes A x;y p x;y ofthestoredmuonsmovecoherentlyperturn N : x ( ;N )= x 0 ( )+ D x ( ) + A x p x ( )cos(2 ˇ x N + ˚ x ) ; y ( ;N )= y 0 ( )+ A y q y ( )cos(2 ˇ y N + ˚ y ) : (3.58) FromtheharmonictransversemotioninEq.( 3.58 ),whichresultsfromthelinearODEs withtheformofHill'sequations 3 ,oscillationsofthemismatchedbeamcentroidsandwidths emerge.AsexplainedinSec. 3.4 ,thecoherentmodulationstendtode-cohereafterbeam injectionwherenonlinearamplitude-andmomentum-dependenttuneshiftsplayat role. Therearetwomechanismsthatdriveclosedorbitdistortionsrelativetotheidealorbit, 3 AsexplainedinSec. 3.4.3 ,nonlinearitiesdrivebetatronamplitudemodulationsforwhichcasesEq.( 3.58 ) doesnotapply. 100 namely,theasymmetricESQvoltageapplicationduringtheturnsafterbeaminjection (seeSec. 3.2.1 )forintentionalcollimationisoneofthemandthepresenceofmagneticand electricsteeringerrors. Intheformercase,theimbalancebetweenvoltagesfromtop/bottomandinner/outer ESQplatesinduceanelectricskewandnormaldipolecomponent,respectively.Theis implementedasconstantsymplectictermsthatactonthetransfermapas(seeEq.( 3.21 )) C a;b = V D e b 0 e l p 0 v 0 ; (3.59) where V = V 0 V 0 isthebetweennominalandmis-poweredvoltages(e.g., V 0 =18 : 3kVand V 0 =13 : 3kVforseveralRun-1datasets), D e =5cmistheESQaperture, and b 0 =0 : 34428isthenormaldipoletermforonetop/bottomESQplateat1Vand theothersat0V[ 89 ].The l termistheESQsegmentlengthoverwhichtheadditional dipoletermisapplied( ˘ 1cmintheimplementation).Oncetheregularone-turnmapis preparedinthismanneratseveralazimuthallocations,their x 0 and y 0 pointsare calculatedwithEq.( 3.36 )whichalltogethercomposetheradialandverticalclosedorbits. Figures 3.23 and 3.24 displaysclosedorbitsduringcollimationinredcolor.Theother mechanismthroughwhichtheidealorbitisdistorted,steeringerrors,aredrivenmostly bytheelectricandmagneticnormal/skewterms(seeSec. 3.2.2 ).Figure 3.25 showsclosed orbitdistortionsduringRun-1,wherethemagneticmultipolecotsweretakenfrom fourtrolleyrunsateachRun-1dataset,named(andorderedinchronologicalorder)as 60h (1a), HK (1b), 9d (1c),and EG (1d).Theradialclosedorbitevolvedduetotemperature- inducedinstabilitiesinthenormalmagneticandsimilarlyfortheverticalcase,whose ontheoverallverticalshiftisdeterminedfrompositronsverticaldataasrecordedby 101 Figure3.23:Radialclosedorbits( =0)at5 s(redcurves),20 s(greencurves),and 1000 s(blackcurves).Ontheleft-sideplotHV=18 : 3kV,whereasHV=20 : 4kVforthe plotontherightside.Theintentionalstretchingoftheorbitduringscrapingincreasesthe probabilityofoutermostmuonstohitacollimatorand,inthisway,minimizemuonloss rates. Figure3.24:Verticalclosedorbits( =0)at5 s(redcurves),20 s(greencurves),and 1000 s(blackcurves).Ontheleft-sideplotHV=18 : 3kV,whereasHV=20 : 4kVforthe plotontherightside.TheinducedskewdipoleESQcreatedfromtheHVimbalance betweenthetop/bottomplatesshiftstheverticalclosedorbitforbeamscraping. 102 Figure3.25:Representativeclosedorbitsduringthedatasetsoftherun(Run-1)ofthe experiment(60h(1a),HK(1b),9d(1c),andEG(1d)).Fluctuationsintemperature thedipoletermsofthemagneticwhichledtotclosedorbitsperRun-1dataset. calorimeters;theazimuthalaverageoftheimplementedskewdipole h B 0 a 0 i istunedto matchthesemeasurements,whereaverticalof1mmcorrespondsto h a 0 iˇ 19 : 5ppm. ItisassumedinthesecalculationsthatfortheRun-1dataset h a 0 iˇ 0,whichrecreates theexpectedcaseofthesurfacecorrectioncoils(SCC)averagingouttheradialeld. 3.4NonlinearBeamDynamics Similartothelinearmapcomponentsdescribedtothispoint,nonlinearmaptermsarealso calculatedbyintegrationoftheODEswherehigherorderrelationsareconsidered[ 72 ].The nonlinearmap M ,computedtoorder n ,transferstheinitialparticlecoordinates ~z ( 1 )at 1 toitsstate ~z ( 2 ): ~z ( 2 )= M (1 ! 2) ~z ( 1 ) : (3.60) Inthisextendedregime,thetransfermap M = fM x ; M a ; M y ; M b ;::: g istreatedasacol- lectionofDAvectorsinCOSYINFINITYandequivalenttothefollowingTaylorpolynomial 103 expansion: z i ( 2 )= X j 1 =0 X j 6 =0 z i x j 1 a j 2 y j 3 b j 4 l j 5 j 6 ( x j 1 a j 2 y j 3 b j 4 l j 5 j 6 )( 1 ) : = C i + 6 X j =1 z i j z j z j ( 1 ) + X j 1 + + j 6 2 z i x j 1 a j 2 y j 3 b j 4 l j 5 j 6 ( x j 1 a j 2 y j 3 b j 4 l j 5 j 6 )( 1 ) : (3.61) Thesummationsover j 1 ;j 2 ;:::;j 6 startatzero(constantterms)andgouptotheorder n =10(i.e.,termswith P 6 i =1 j i 10)inthepresentedanalysis,whichisfoundtobe ttocapturealltherelevanthigh-ordercontributionsfornonlinearsuchas the20polefromtheESQelectrostatic 4 Nonlinearcontributions O (2)arecaptured inthetermsofthelastlineofEq.( 3.61 );duetotheirrelativelysmalleronthe muonsmotionwhentlyawayfrombetatronresonances,theycanbetreatedas aberrationsorcorrectionstothelinearsystemofthe g -2storagering.Nevertheless,the understandingofexperimentalobservationsandthehighlevelofprecisiondemandedfor theE989measurementrequiresthecharacterizationofthenonlinearmotion.Inparticular, nonlineardetuning,betatronresonances,andnonlinear-drivenmuonlossesareelaborated next. 3.4.1Momentum-andamplitude-dependenttuneshifts Fromdesign,thefurthermuonsmoveawayfromtheclosedorbit,andthemoretheydeviate fromthenominalmomentum,thestrongertheofguidenonlinearcomponents ontheirdynamicsbecomes.ThiscanbeseeninEq.( 3.61 );as z i !1 ,thenonlinearpoly- 4 NonlineartermsarealsoincludedinEq.( 3.37 )andinanyre-expansionofmapsaroundpoints. 104 nomialcotsgainanincreasingweightinthetransformationof ~z ( 1 ).Thus,betatron tunesarenolongersubjecttoonlytheusualfromlinearforces.TheCOSY-based environment,COSYINFINITY,allowstocapturetheofnonlinearitiesonbetatron tunesviaitstialalgebranormalform(NF)algorithm;forathoroughexplanationof thealgorithm,see[ 72 ].Inessence,thealgorithmyieldsasetofNFcoordinatesforwhich thetransformedmapisgreatlyandleadstorotationalinvarianceuptothecalcu- lationorderintheNFspace.Fromthisorder-by-ordertransformation,theNFcoordinates f x;a;y;b g!f s 1 ;t 1 ;s 2 ;t 2 g exhibitcircularbehaviorsthatdependonlyonamplitudesand parameterssuchas : 0 B @ s 1 t 1 1 C A 2 = 0 B @ cos( x ( r 1 ;r 2 ; )) sin( x ( r 1 ;r 2 ; )) sin( x ( r 1 ;r 2 ; ))cos( x ( r 1 ;r 2 ; )) 1 C A 0 B @ s 1 t 1 1 C A 1 ; (3.62) where r 2 1 = s 2 1 + t 2 1 and r 2 2 = s 2 2 + t 2 2 arethesquaredamplitudesinnormalform.Thus, withtheNFmapandastraightforwardcomputationof,say,thearccosinewithintheDA framework,thefulltunesareexpressedasTaylorexpansionsuptotheorderoftheoriginal map: x = X i;j;k =0 x j r i 1 r j 2 k r i 1 r j 2 k ; y = X i;j;k =0 x j r i 1 r j 2 k r i 1 r j 2 k : (3.63) Forexample,nominaltunesfromthelinear,origin-preservingmapscorrespondtothe x;y j 1 termsandlinearchromaticitiesarenaturallycalculatedfrom ˘ x;y = x;y j . AcaveatoftherotationalinvarianceinEq.( 3.62 )emergesforsystemsnearthee ofresonances.Forthesecases,theNFalgorithmcannotremovesomenonlinearmapcom- ponents,andadirectcomputationofamplitude-dependenttuneshiftsisnotpossible.As 105 such,amplitudesofraysundertheoftheresonance-driventermsarenotconstant and,thus,tunescanevolveoverthemotioninphasespace,losingtheirphysicalmeaning; however,theirlimitingaveragewouldcorrespondtotheformalofatuneasinreg- ularphasespace.Animportantconsequenceisamplitudemodulationsthatincreasemuon lossratesinthemuon g -2experiment,asdescribedinSec. 3.4.2 and 3.4.3 . Athoroughinvestigationofmomentum-andamplitude-dependenttuneshiftsforthe muon g -2storageringisreportedin[ 90 ].Ofspecialimportanceonnonlineardetuningisthe 20polefromtheESQelectrostaticpotential.Itspresenceinthestorageregionismanifested intheamplitude-dependenttuneshiftsasshowninFigs. 3.26 and 3.27 (for =0). Theshiftscalinggrowsinan8th-powerfashionfromtheclosedorbittowardthelimiting apertures,asaconsequenceoftheelectric20pole[ 90 ].Ontheotherhand,magnetic imperfectionsdonotcontributenoticeablytotuneshifts,asindicatedinthe Inthemomentum-dependentcase( r 1 ; 2 =0),tuneshiftsaremostlyproportionalto thelinearchromaticitiesfor j j < 0 : 2%(seeFig. 3.28 ).Beyondthatregionandwithin themomentumacceptance,tunesexperienceshiftsof ˘ 4 10 3 ,beingnegativeinthe verticalcase.Theppm-levelmagneticerrorsdonotcontributenoticeablytomomentum- dependenttuneshifts. Inreality,muonswiththesamebetatronamplitudesbutseparatedinmomentum(see Eq.( 3.58 )foragoodapproximation)scantregionsofthestoragevolumeand,con- sequently,tnonlinearities.Thus,momentum-andamplitude-dependenttune shiftshavetobetreatedsimultaneouslyasindicatedby r i crossedtermsinTable 3.3 . Particle-by-particletunespreadinginacceleratormachinescanleadtobeamde-coherence [ 91 ].The g -2ringisnotanexception.BeamtrackingsimulationswiththeCOSY-based modelindicateastrongofnonlinearitiesonthebeamde-coherenceofthecoherent 106 Figure3.26:Amplitude-dependenttuneshifts( =0)withinthestorageregionwithout magneticimperfections(HV=18 : 3kV).Thehorizontalaxiscorrespondstoradial betatronamplitudesandtheverticalaxisrepresentsverticalamplitudes. Figure3.27:Amplitude-dependenttuneshifts( =0)withinthestorageregionwithmag- neticimperfections(HV=18 : 3kV).Thehorizontalaxiscorrespondstoradialbetatron amplitudesandtheverticalaxisrepresentsverticalamplitudes. 107 Figure3.28:Momentum-dependenttuneshifts(nobetatronamplitudes)withandwithout magneticimperfections(HV=18 : 3kV).Nonlinearshiftstakeplacefor j j > 0 : 2%, insidethemomentumacceptance. betatronoscillation(CBO).Furthermore,toyMonteCarlosimulations[ 92 , 93 ]withonly COSY-basedtuneshiftsarecapableofreproducingtheobservedbeamde-coherence.Asan example,bytakingmomentum-andamplitude-dependenttunesinTable 3.3 (uptoorder10) withmomentumandamplitude 'sindependentlyandscaled,andreproducingasimulated linearmotionasinEq.( 3.58 ),theradialde-coherencerecordedbythe g -2strawtracking detectorsiscloselyrecreated(seeFig. 3.29 ).Withouttuneshifts,thebeamstillreduces itsinitialCBOduetothelongitudinalrecombinationbetweenhigh-andlow-momentum muons,asshowninFig. 3.47 .However,withoutthenonlineardetuning,observationsare notreproduced.Attheendofthemeasurementperiod( ˘ 700 safterbeaminjection),a CBOamplituderemanentofabout1mmmightsurviveassuggestedbythesesimulations extendedintime(seeFig. 3.47 ).Also,Fig. 3.47 suggestssmallre-coherencepulsationsmight manifestbuttoalowextent. 108 Table3.3:TunecotsatHV=18 : 3kVuptoorder,normalizedby100 order .For example,( x j )= 0 : 13199935. Radialtune x j r i 1 r j 2 k order ijk 9.4446263E-01 0 000 -1.3199935E-03 1 001 -1.3028752E-06 2 200 4.2332676E-06 2 020 -3.9055732E-05 2 002 6.9991197E-05 3 201 -4.2045086E-04 3 021 7.5544892E-04 3 003 2.3798658E-05 4 400 -4.1727784E-04 4 220 6.0854702E-04 4 040 2.2570512E-03 4 202 -1.3199217E-02 4 022 1.1914645E-02 4 004 1.9657417E-06 5 401 -3.3974259E-05 5 221 4.7589424E-05 5 041 7.9046305E-05 5 203 -2.8914099E-04 5 023 3.0339641E-04 5 005 Verticaltune y j r i 1 r j 2 k order ijk 3.3081444E-01 0 000 3.8975399E-03 1 001 4.2332682E-06 2 200 5.0651491E-06 2 020 4.3361598E-05 2 002 -4.2045082E-04 3 201 6.3490582E-04 3 021 -2.2468635E-03 3 003 -2.0863889E-04 4 400 1.2170941E-03 4 220 -5.9016314E-04 4 040 -1.3199216E-02 4 202 1.9295418E-02 4 022 -3.4840049E-02 4 004 -1.6986818E-05 5 401 9.5177294E-05 5 221 -5.3892581E-05 5 041 -2.8913252E-04 5 203 4.1910614E-04 5 023 -4.7759203E-04 5 005 3.4.2Betatronresonances Whenthetransversemotionofmuonscoupleswiththeperiodicityoflocalforcesinthe storagering,theirbetatronoscillationamplitudesmayrapidlydivergefromthestorage volumebythecollimators,ormorecommonlyexperienceturnbyturnamplitude modulationsinawaythattheyenduphittingthecollimatorsduringdatataking[ 69 ].Such muons,commonlyknownas lostmuons withinthemuon g -2collaboration,giveupenergyto thematerialtheyinteractwith(seeSec. 3.2.4 ),movinginwardandwhosedaughterpositrons escapefromdetection;inSec. 3.4.3 theimpactofsuchmuonsonthe ! a measurement isexplained.Theelectrostaticscrapingrightafterbeaminjectioniscrucialforremoving 109 Figure3.29:Beamradialde-coherencewith(left)andwithout(right)tuneshiftsatthe readoutlocationofthestrawtrackingdetector(station12).Asareference,datafromthe trackerisshowninred.Thetwoplotsontopdisplayradialbeamcentroids,whereasthe bottomplotscorrespondtoradialbeamwidths. muonswithlargebetatronamplitudes.However,evenmuonswithinitiallysmallbetatron amplitudesaroundunstablemotionregionsandnonlineardpointsinphase-spacecould becomelostatlatertimesunderthemodulationclosetoresonantconditions. High-statisticalnumericalstudiesofmuonlossrateswereperformedwiththeCOSY- basedmodeltounveilthesetofESQoperatingpointswheretbetatronresonances occur.InFig. 3.30 ,simulationresultsforseveralhigh-voltagevaluesappliedtotheESQare presented.Theshowsthefractionoflostmuonsduringthetimeinterval121 186 s 110 Figure3.30:Fractionofmuonlossesbetween121 186 safterbeaminjectionwiththe COSY-basedmodelforseveralESQMeasurements[ 94 ]|showninarbitrary units|resultfromdetectionsofminimumionizingparticlesthatdeposit ˘ 170MeVof energyintwoadjacentcalorimeterswhiletwocollimatorswereinsertedtothestorageregion. afterbeaminjection.Verticalerrorbarscorrespondtothestandarderrorofthemultiple simulationsexecutedinparallelthatwerenecessarytosurpasstheresolutionrequiredand clearlydistinguishresonancepeaks(i.e., ˘ 6 : 8 10 6 muonsatbeaminjection). DuetothesensitivityofE989,eventherelativelysmallofhigh-orderresonance conditionsareavoided.Todescribetheinterplaybetweentheimperfectionsinthe storageringandtheresonantconditions,theringisidealizedasasuperpositionofauniform verticalmagneticandelectricquadrupolecomingfromtheESQ,whereallthe extrafromtheESQdesignandmagneticimperfectionsaretreatedasperturbations tothelinearcase.Inparticular,forsimplicitythismodelassumesasmoothbehaviorofthe unperturbedbetatronoscillationsthatdoesresembletherealcase[ 95 ]: x ( ) ˇ H cos(2 ˇ x + ˚ x ) ;y ( ) ˇ V cos(2 ˇ y + ˚ y )(3.64) where istheazimuth,and H and V areconstantradialandverticalbetatronamplitudes(a 111 goodapproximationintheweaklyfocused g -2storedmuonbeam),respectively.Considering bothmagneticandelectricpotentialsintheform x;y; )= 1 X l =0 1 X m =0 1 X N =0 C l;m;N x l y m cos( N + ˚ lmN ) ; (3.65) Eqs.( 3.64 )and( 3.65 )canbecombinedtoestimatethetransversalforcesintermsof2k-pole terms( k = l + m )and N thazimuthalharmonics.TheleadingtermsfromFourieranalysis oftheperturbationsyieldthefollowingconditionsthatresonatewiththebetatronmotion: ( l 1) x y N = ;l x ( m 1) y N =(3.66) where= x;y dependingonthethatdrivestheresonance.Ingeneral,vertical forcesinducemoremuonlossesthanhorizontalforcesduetotherelativelysmallervertical admittanceofthering(seeSec. 3.5.1 ).Thestorageringaremostlyfour-foldsymmetric, thusfavoring N =0 ; 4 ; 8 ;::: .Figure 3.31 depictsresonancelinesaswellastheoperating setpointsofthestoragering.Resonanttunelinesupto k =10areshown,whichare tforourpurposesofcoveringallthemainmultipoletermsthatareimplementedin theCOSY-basedringmodel,includingtheESQ20pole. AlongwiththeresonanceconditionsshowninFig. 3.31 ,theobservedresonancesinthe storageringdependontheangleadvancementspreadinphase-space,betatronamplitudes, momentumdeviations,andinitialphaseofmuonsrevolvingthestoragering.Thecom- binationofthesepropertiesdictateshowfrequentlyamuoncrossesunstableconditionsin phase-spacetoexperienceresonantornot.Forinstance,thecontinualgrowthof resonancepeakwidthsforincreasingHVobservedinFig. 3.30 relatestotheverticalangle 112 Figure3.31:Anillustrationofresonanttunelinesandoperatingpoints(i.e.,ESQnomi- nalvoltages)ofthe g -2storagering.Thehigh-voltageappliedtotheESQplatestoset operatingpointsareshownnexttoresonancespredictedbynumericalstudieswiththe COSY-basedringmodel.Measurementsoflostmuonshaverevealedresonancepeaksat ˘ 13 : 1 ; 16 : 8 ; 18 : 6 ; and21 : 1kVaswell(seeFig. 3.30 ).Lostmuonsmeasurementshavenot beenperformedforhigherESQvoltages.Redmarkersshowtheoperatingpointsusedin Run-1. 113 advancementspread.Itisworthnotingthatthisangle,whichcanbedeastheturn- by-turnangularevolutionofthemuoncoordinatesinverticalphase-spaceatanarbitrary azimuthlocation,advancesinamoreperiodicmannerastheverticalbetatronfunctionof thestoragering, y ,tlydecreasesonaverageforhighervoltageswithintheexplored HVrange.Regardingbetatronamplitudesandmomenta,theseproperties,whichareunique toeachmuonwithinthestoredbeam,indirectlytheattributesofthepeakspresented inFig. 3.30 aswell,sinceresonanceconditionsinEq.( 3.66 )arenotuniquelybyall thestoredmuonsduetononlineartuneshiftsinthe g -2storagering.Thetunefootprintfor HV=18 : 3kV(usedinRuns1aand1d)inFigure 3.32 illustratesthisEventhough theoperationalsetpointisnotatresonance,itiscloseenoughtothe3 y =1resonantcondi- tionsothatmomentum-andamplitude-dependenttuneshiftspushmuonstotheresonance region. Tominimizesystematicerrorsduetomuonlosses,thestorageringmustrunatoperating pointstlyawayfrombetatronresonances.AsshowninFig. 3.31 ,thesystemfallsinto resonancearoundspsetpointsandnotateveryintersectionwithresonanttunelines. Thisfeatureismostlyduetotheratherspsetofnonlinearelectrostaticmultipoles providedbythe4-foldsymmetricESQ,wherethe20polehasthelargeststrength.In addition,thereinforcementofmultipleresonantconditionsisalsoamaindeterminantof theobservedandmeasuredbetatronresonances,whichcanbenoticedintheirproximity tothepointswhereseveralresonanceconditionsconverge.Nevertheless,eventhoughthe multipolesthatdescribethemeasuredmagneticimperfectionsdonotfavorparticular azimuthalharmonicsduetotheirsomewhatrandomdistributionalongtheringazimuth, theystillhaveanimportantonbetatronresonances,asshownnext. 114 Figure3.32:WiththetuneshiftsreadilyavailablefromtheCOSY-model,momentumand normalformradiiofstoredmuonsfromarealisticinitialdistributionasdescribedinSec. 3.2.5 areusedforthecalculationofthetunefootprint.TheESQsetpointis18 : 3kV.Dueto nonlinearamplitude-andmomentum-tuneshifts,aconsiderablefractionofthestoredmuons isbytheresonancearoundintunespace,especiallythe3 y =1resonantcondition drivenbytheskewmagneticsextupoleterm.Red(blue)markersindicatehigh(low)density ofentriesinthetunefootprint. ResonanceDrivingTerms Adeepunderstandingoftheforcesthatdrivebetatronresonancesistantamounttoade- tailedcharacterizationoftheconditionsinthestorageringthatexcitemuonlossrates.By identifyingalltheresonancesthatemergefromtheimperfectionsoftheringalreadyknown 115 (i.e.,nonlinearelectricperturbationscontributedbytheESQandmagneticinhomo- geneitiesfromNMRprobesmeasurements),itispossibletoaddresstheoriginofresonant peaksfromlostmuonsmeasurementsnotaccountedforintheCOSY-basedmodel.Extra peakscouldbegeneratedbymis-poweringormisalignmentsoftheESQplates,transient magneticoutsidethebandwidthoftheNMRprobes,andotherfactorsnotincludedin thesymplectictrackingsimulationsthatotherwisewouldbetoevaluateandwould alsodirectlyttheevolutionof ! a undermomentum-spincorrelations. Inaddition,motothedefaultsettingsofthesimulatedringmodelpermit determiningthedrivingtermsthatgenerateeachbetatronresonancepeak.InFig. 3.33 , simulationresultsareshownforthefollowingmototheoriginalringmodelcon- (seemarkerlegendinFig. 3.33 foreachcase): (i) AllE-multipoles,inhomogeneitiesON(originalcase,red) (ii) Only4thE-multipole,inhomogeneitiesON(gold) (iii) No20thE-multipole,inhomogeneitiesON(magenta) (iv) AllE-multipoles,inhomogeneitiesOFF(blue) (v) Only4thand20thE-multipoles,inhomogeneitiesOFF(green) (vi) No20thE-multipole,inhomogeneitiesOFF(orange). The2DmultipoletothemagneticNMRmeasurements(abbreviatedas\B- inhomogeneities"inthelistabove)introducelostmuonfractionsof ˘ 6 10 4 ˝ evenatoperatingpointsawayfromthemainbetatronexcitations.Betatronresonance peaksobservedfrommeasurementsand/orsimulationscanbeexplainedfromresonance 116 Figure3.33:Fractionofmuonlossesfromsimulationsduringthetimeinterval121 186 s afterbeaminjection.detailsarelistedonpage 116 .Thebottom depictslostmuonfractionswithareducedverticalrangetodiscernlosseswhenmagnetic imperfectionsarenotaccountedfor. 117 conditions(Eq.( 3.66 )),whichcanbeindependentlydrivenbyoneorseveralelectricand magnetichigh-ordermultipolesinthestorageringmodel(seeTable 3.4 ). Table3.4:Resonantconditionsrelatedtobetatronexcitations. HV [kV] x + y = c 13.13 x +4 y =4 16.8 x 3 y =0 18.63 y =1 21.12 x +6 y =4 24.7 x +8 y =4 30.43 x +3 y =4 Nevertheless,Fig. 3.33 showstheinterplaybetweenmagnetic(electric)nonlinearitiesand electric(magnetic)resonances.Forinstance,foraringwith HV ˘ 13 : 1kVtheresonance drivenbytheelectric20-poletermsderivedfromthecurvatureoftheplatesisboostedby lowerordernonlinearitiesfromtheimperfectmagneticseveralresonanttunelinesnear suchanoperatingpointreinforcetheresonance.However,byanalyzingthehigh-orderterms ofthestorageringtransfermapinNormalForm(NF)itispossibletounderstandwithmore detailtheroleofnonlinearitiesinrelationtoresonances. For HV ˘ 16 : 8kV,eitherthemagneticorelectricguidecandrivetheresonance condition x 3 y =0.Themagneticforcesbuildupalargerresonancethantheelectric case,asfoundinsimulations.Intheelectriccase,electrostaticpotentialcoetsfrom the20poleareneededtoseearesonance.Ontheotherhand,themagneticoctupolecan drivetheresonancethroughverticalforcesfromhorizontalmagneticproportionalto B x / xy 2 .Azimuthalindependenceoftheelectric/magneticpotential(i.e., N =0)is favoredforthisresonance. The HV ˘ 18 : 6kVresonancedeservesparticularattention[ 96 { 98 ]asitisnearenoughthe ESQoperationalsetpointsduringRuns1aand1d(i.e.,18 : 3kV).SimulationswiththeCOSY- 118 Figure3.34:Ontheleft,magneticskewsextupolecot a 2 asmeasuredbyNMR probes.Ontherightplot,Fourierdecompositionofthecot a 2 inazimuthalharmonics a 2 = P n N =0 a 2 ;N cos( N + ˚ N ).The N =1term,whichisthemaindriveroftheresonant condition3 y =1,isdepictedingreencolor. basedmodelindicatethattheresonanceispurelydrivenbymagneticinhomogeneities, thoughareciprocitybetweenresonancesandnonlinearitiesfromtorigin|electricor magnetic|exists.Thedrivingforcebehindthisverticalresonanceistheradialmagnetic B x ( x;y; )= B 0 a 2 ;N =1 cos( + 1 ) y 2 ; (3.67) Thecot a 2 ;N =1 istheazimuthalharmonicofthemagneticskewsextupole;afast Fouriertransformation(FFT)ofthemeasured a 2 inFig. 3.34 (rightplot)showsanon-zero N =1component,tlylargetodriveanonlinearbetatronresonanceasdepictedin Fig. 3.35 . Theofthe3 y =1resonanceextendsallthewaytotheRun-1ESQsetpoint HV =18 : 3kVduetoamplitude-andmomentum-dependenttuneshifts.Byturningonand theskewsextupolecomponentofthemagneticimplementationintheCOSY-based model[ 98 ],itwasdemonstratedthatintheabsenceof a 2 thebetatronresonancepeakat HV ˇ 18 : 6kVdoesnotbuildup.Moreover,itwasalsofoundthatbyreducingthemost 119 Figure3.35:Lostmuonsfractionoveramuonlifetimeafterthescrapingstageiscompleted. Inred(triangularmarkers),allthemagneticmultipoletermsareturnedonduringthebeam trackingsimulation,whereasinblue(diamondmarkers),theseareturnedInsubsequent simulations,itwasfoundthatonlythemagneticskewsextupoletermfrommeasured inhomogeneitieswasttodrivethebetatron-resonancepeakat HV ˇ 18 : 6kV. prominentpeakoftheskewsextupolecomponentaroundtheazimuthnoticeablyreduces a 2 ;N =1 and,consequently,theresonanceHowever,itmightbemorepracticalto theESQsetpointawayfrom HV ˇ 18 : 6kVratherthansmoothingouttheskew sextupolemagneticinhomogeneitieswithactiveshimming,butitdependsonthesp operationalcircumstances.Theimpactofequivalentelectricskewsextupolecotsfrom ESQplatemisalignmentswasalsoanalyzed.Qualitatively,thesharpnessoftheobserved muonlossrateresonanceat HV ˇ 18 : 6kVwasnotreproducedinthisscenario.And quantitatively,themisalignmentnecessarytorecreateadiscernibleresonance peakwasunrealisticbasedonsubsequentmisalignmentsurveys[ 99 ].Thus,itwasconcluded thatthemagneticskewsextupolewasthemaindrivingforceofthe3 y =1resonance. 120 3.4.3Lostmuonmechanisms Forastoredbeamwithmomentum-phasecorrelationsatinjectiontime,muonsthatperma- nentlyescapefromthestoragevolumeduringdatataking(andwhosedecaypositronsarenot detectedbythecalorimeters)canpotentiallybiasthemeasured ! a byinducingslowdrifts inthephaseinEq.( 1.25 ).SincetheBeamDeliverySystem(BDS)deliversamuonbeam withacorrelationbetweenthe g -2phaseandmomentumofabout10mradper =0 : 01 (Sec. 2.3.4 ),areductionoftheseso-called lostmuons istantamounttoamitigationin ! a systematicerrors. BytrackingsymplecticallythemuonbeamwiththeCOSY-basedmodelfor n =1and n =10ordertransfermaps,adetailedpictureofthemechanismsthatdrivemuonstobelost canbedeveloped.Externallytotheworkpresentedinthisdocument,severalmechanisms thatpotentiallydrivemuonlosshavebeenstudied(fordetails,see[ 100 ]),amongthemmuon scatteringwiththeresidualvacuuminthestorageregion,transientelectromagnetic andnonlinearawayfromresonances.Intheanalysis,itwasalsodeterminedthatthe mainreasontherearelostmuonsduringproductionmeasurementsinthelinearregimeis thescrapingmethodimplementedafterbeaminjection;asmallfractionofmuonsbeyond thelimitingaperturesofthestorageregion(i.e., p x 2 + y 2 > 45mmatcollimatorlocations) canrevolvetheringforthousandsofturnsbeforeadirectcollisionwithacollimator.Linear simulationsareabletoreproducethemuonlossratesshowninFig. 3.30 awayfrombetatron resonances[ 100 ]. AtHV=18 : 3kV(thesetpointusedinRuns1aand2d,andnexttotheESQvoltageHV= 18 : 2kVusedinposterior g -2runs),nonlinearresonancesdrivenbythe3 y =1condition playatroleinmuonlossrates.Asshownnext,undertheRun-1injectionsettings 121 nonlinearlossratescontributetoconsiderablyhighermuonlossratesthanlinearlydriven losses. Itishelpfultostartwithamuonthatdoesnotgetlostinthelinearregimeandtakeits motionasareference,whereHV=18 : 3kV.Inthestroboscopicview(i.e.,atoneazimuthal location),suchmuonfollowswellipticalmotioninradial/verticalphasespaceas indicatedbyEq.( 3.58 )andexpectedfromLiouville'stheorem.Also,itsradiiinnormalform coordinatesareinvariant.Figure 3.36 illustratesthemotioninthiscase.Consequently,the Figure3.36:Phasespacecoordinatesandmaximumexcursionofamuonnotlost(reference case). maximumtransverseexcursion r 0 relativetotheidealorbitcanbedescribedas r max = q ( A x + j D x + x 0 j ) 2 + A y + j y 0 j 2 : (3.68) Infact,thisdynamicalevolutionisalsoexpectedinthenonlinearregimeonlywhenthere arenoresonance-driventermsthatcouldthepursuedrotationalinvarianceofanormal formtransformation(seeEq.( 3.62 )). LostmuonsalsoemergeinCOSY-basedlinearsimulations.Eventhoughamuonbeam withdata-driveninitialconditionsat4 s(seeSec. 3.6.3 )andinitiallycleanedviaconven- 122 tionalscraping(seeSec. 3.2.1 )istrackeddown,alas,about0 : 2%ofthestoredmuonsat t> 30 swith r max >r 0 arenotremovedfromthebeam.Figure 3.37 showsatypical sampleofthesemuonsandFig. 3.38 theirmaximumexcursions,whichtheirsur- vivalforseveralmicrosecondsinsidethedata-takingtimerange.High-orderresonances Figure3.37:Trajectoriesinphasespaceofthreetmuons(distinguishedbycolor)in thelinearregime. areirrelevantinthelinearregime,andasaconsequencelinearchromaticitiesaretheonly contributorstosmalltuneshifts,giventheconstrainedmomentumacceptanceof ˘ 0 : 55% (seeFig. 3.39 ). 5 Nowtheattentionisturnedtothecasewherenonlinearmotionisaccountedfor.The sameinitialmuondistributionandscrapingschemeareusedasinthelinearsimulations 5 Inthissubsection,themeaningofcolorcodingissharedin 123 Figure3.38:Maximumexcursionsofthreelostmuonsinthelinearregime. describedabove.However,thesymplecticconditionisenforcedviaasymplectictransfor- mationofthetransfermapasdescribedin[ 70 ].Inthismanner,theJacobianofthemap isequaltoone,andconsequentlythephasespacevolumeisconservedturnbyturn,an essentialcharacteristicofHamiltonianmotion. Figure 3.40 showsphasespaceinregularcoordinatesofthreemuonslostinthenonlinear, symplecticsimulations.Theradialmotionisnotbyresonanceterms,inagree- mentwiththe3 y =1verticalresonancedrivenbyskewsextupoletermsofthemagnetic Ontheotherhand,verticalamplitudemodulationsmanifestand,asaresult,max- imumexcursionsevolveovertime(seeFig. 3.41 ).Inthelinearmechanism,thestatistics involvedinthetimeittakesamuonwith r max >r 0 isplain.Thehigher r max is,thefaster themuonislost(lossratescanbederivedfromEq.( 3.58 )).Ontheotherhand,muons nonlinearlymodulatedwhosemaximumexcursionsintermittentlyvisitvaluesbeyondcolli- mationaperturessurviveforlongerperiods,asindicatedbylinearversusnonlinearmuon losssimulationcomparisons(seeFig. 3.43 ). Whentheshiftedtunesofnonlinearlydrivenlostmuonsareextractedasexplainedin 124 Figure3.39:Tunesofthreelostmuonsinthelinearregime.Notetheproximitybetweenthe redandblackentries(theredmarkerpartiallycoversthemarkerinblackcolor). Sec. 3.4.1 ,alltheircoordinatesintunespacefallwithinthevicinityofthe3 y =1resonance asshowninFig. 3.42 ;astheymovearoundunstableandstable3-periodpointsin phasespace[ 101 ],tunesconstantlyshiftinconjunctionwithverticalbetatronamplitudesin phasespace. 6 Theperiod-3pointstructuresareclearlyvisiblefrommuonsymplectic trackingwithinthestorageregionandmomentumacceptance,asshowninFig. 3.44 .Vertical betatronamplitudesaremodulatedasmuonsmoveinphasespacearoundthestableand unstablepointsgeneratedbythe3 y =1nonlinearresonance.Althoughthemotionis 6 Animatedexamplesoflostmuonshiftingtunescanbefoundin[ 102 ]. 125 Figure3.40:Trajectoriesinphasespaceofthreetmuons(distinguishedbycolor)in thenonlinearregime. stable,modulationsinterferewiththetimeittakesforamuontobecomelost. FurtherstudiessummarizedinFig. 3.45 wereaimedatunderstandingthemulti-dimensional natureofmuonlossrates.UsinganinitialdistributionbasedonstrawtrackersRun-1adata andthescrapingschemeusedduringRun-1,itisconcludedthatmuonlossratesarechar- acterizednotonlybyresonance-drivenbutbythenumberofinsertedcollimators.In thepresentedresultsregardingmuonlossdrivingmechanisms,twocollimatorswereinserted (downstreamofESQstationsQ3LandQ4S)asinRun-1duringtrackingsimulations.For posteriorruns,allcollimatorsareinserted,forwhichcasemuonlossratesaresubstantially reducedassuggestedbytheCOSY-basedmodelsimulations.However,anotherfactorto accountforisthebeaminjectionqualitywhichdeterminesthenumberofoutermostmuons 126 Figure3.41:Maximumexcursionsofthreelostmuonsinthenonlinearregime. likelytogetlost.Inaddition,Fig. 3.45 alsodisplaysacomparisonwithandwithoutthe ofthetime-evolvingESQduringRun-1d(seeSec 3.6 ),indicatinghigheramuon lossfractionintheformercaseasthenominaltunemovestowardthe3 y =1resonance andtheverticalbeamcentroiddriftsoverthedatatakingperiod.Moreimportantfora reliableanalysisoflostmuonsisthesymplectictransformationfortracking;initsabsence, anoverallemittancegrowthisobservedinsimulationresults,consequentlyproducingan muonlossfractionincrease(seeFig. 3.45 ). 3.5NominalCharacteristicsoftheStoredBeam Undernormalcircumstances,thebeamdeliveryandinjectionprocessintothe g -2storage ringatFermilab,aswellastheelectric/magneticlatticeinsidethering,specifythetempo- ralandazimuthalbehaviorofthemuonbeam.Duetothebeamtransversebeing mismatchedwiththeringopticalsettings(especiallyinradialphasespace),transverseos- 127 Figure3.42:Tunesofthreemuonsinthenonlinearregimebeforegettinglost. cillationsofthebeamcentroidsandwidthsarecommon. 7 Ontheazimuthalside,beam centroidstypicallyfollowmm-levelclosedorbitsandbeamradial/verticalwidthsaremod- ulatedbythevariationsoftheopticallattice(ofabout0 : 4mmalongtheringazimuth)in analmostfour-foldsymmetricpattern.Bytrackingdownarealisticbeam|whoseinitial conditionsarepreparedbasedonRun-1abeamdata(seeSec. 3.6.3 )|withtheCOSY-based modelundernominalsettingsatHV=18 : 3kV(andmagneticasmeasuredaround Run-1a),representativesamplesofthebeamovertimeandazimutharepresentedinthis 7 Atthetimeofthisdissertation,advancedRFtechniquestosubstituteconventionalESQscrapingwere beingdevelopedbyMuon g -2collaboratorstomitigatebeambeating[ 103 ]. 128 Figure3.43:Muonlossfractionswithnonlinearities(blackcurve)andwithoutthem(gray curve)fromsymplectictracking,COSY-based,beamsimulations.TheESQsetpointwas 18.3kV,twocollimatorswereinserted,andbothcasesstartwiththesamedata-basedinitial distribution.Muonlossratesarehighlyedbynonlinearitiesoftheguide Figure3.44:Stroboscopictrackingintheverticalphasespaceillustratingorbitbehaviorwith twoperiod-3pointstructurespresent[ 101 ],forESQvoltageat18.3kV.Trajectories inblueandgreencolorsareexamplesofmuons(withintheringadmittanceandmomen- tumacceptance)withhighlymodulatedverticalamplitudes.PicturebycourtesyofAdrian Weisskopf. 129 Figure3.45:Muonlossratesfromseveraltrackingsimulations(HV=18.3kV).Theectof damagedresistorsandthefromsymplecticenforcementduringtrackingisshown. section.SeveralexperimentalimprovementsweremadeafterRun-1concluded,wherethe radialCBOamplitudeswerereducedandmomentumspreadbecamebettercentered,reduc- inginthiswaydispersiveHowever,themainfeaturesdiscussedinthissectionare alwaysexpectedtoanobservationallevel. Theresultspresentednextservetocharacterizethetransversefrequenciesofthemuon beamandvalidatetheimplementationofopticallatticefunctions|aswellasadistinction oftheirlimitations|todescribethetransversebeamintensityaroundtheazimuthofthe storagering.Thesevalidationsareimportantsincetheopticallatticefunctionsfromthe COSY-basedmodelareextensivelyusedfor ! a systematiccorrections(i.e.,pitch[ 104 ],phase acceptance[ 44 ],andRun-1[ 105 ]corrections)andthemuonweightingofthemagnetic ~ B [ 106 ]. First,itisusefultospecifytheboundariesofthestoredmuonbeam.Andattheend ofthesection,theinitialmomentum-timecorrelationofthebeaminthemuon g -2experi- mentobtainedfromthesestudieswasusedtodiscoverandestimatethelargestsystematic uncertaintyofthecorrection. 130 Figure3.46:Qualitativecomparisonbetweenstoredmuonratesfromsimulations(gray)at t =186 safterbeaminjectionandmeasuredrelativepositronrates(arbitraryunits)from storedmuondecays[ 107 ].Errorbarscorrespondtostandarderrorsofthemultiplenumerical analysesforeachESQvoltageAstheverticalandradialadmittancesincrease anddecrease,respectively,proportionaltotheESQvoltage,thefractionofstoredmuons increases,reachingaplateauat22kVwherethestoredfractionstartstobecomemore sensitivetotheradialadmittance.Thelowfractionofstoredmuonsisaconsequenceof themomentumspreadbeingabouttwotimeslargerthanthemomentumacceptance,the dispersionmismatchbetweentheendoftheM5-lineandthestoragering,andtheimperfect kickspurposedtoinjectthebeam. 3.5.1Beamboundaries The r 0 =45mmradiusapertureofinsertedcollimatorstogetherwiththeopticallattice oftheringthelargesttransverseemittance(knownasadmittance)andmomentum acceptanceinthestoragering.Fromtheseboundparameters,themaximumbetatronam- plitudes,maximumangles a and b ,andlimitingmomentumcanbeestablishedalongthe azimuth.Furthermore,thestoredmuonsfractionisalsoectedbythesequantities,as showninFig. 3.46 . BythelargestbetafunctionsatcollimatorlocationsandusingEq.( 3.55 ),admit- 131 tancesareas " max x;y = r 2 0 max x;y : (3.69) ForRun-1datasets,theirnominalvaluesareshowninTable 3.5 .Itisworthhighlighting theroleofthemagneticeldinthelargerradialadmittance. Table3.5:StorageringnominaladmittancesforRun-1ESQsetpoints. HV[kV] " max x [ ˇ mm : mrad] " max y [ ˇ mm : mrad] 18.3 268.75792.8262 20.4 266.94697.8427 Thecollimatorgeometryiscircularandnotsquared;thus,admittancevaluesasshownin Table 3.5 wouldcorrespondtoabeamwithverylowintensity. Maximumanglesareobtainedvia a max ( )= p " max x x ( ) ;b max ( )= q " max y y ( )(3.70) andmaximumbetatronamplitudesaregivenby x max ( )= p " max x x ( ) ;y max ( )= q " max y y ( )(3.71) ForanumericalreferenceseeTable 3.6 ,whichdisplaysthesmallvariationsthankstothe relativelyweakfocusingscheme. Table3.6:MaximumvaluerangesinthestorageringalongtheazimuthforRun-1settings. HV[kV] x max [mm] a max [mrad] y max [mm] b max [mrad] 18.3 f 44 : 46 ; 45 : 41 gf 5 : 91 ; 6 : 04 gf 44 : 27 ; 45 : 10 gf 2 : 06 ; 2 : 10 g 20.4 f 44 : 42 ; 45 : 57 gf 5 : 86 ; 6 : 01 gf 44 : 18 ; 45 : 12 gf 2 : 17 ; 2 : 21 g 132 Themomentumacceptanceisalsodirectlycalculatedintheopticalview,wherethe maximumradialdispersionatcollimatorlocationsistaken,whichyields max;min = r 0 D max x ˇ 0 : 56% : (3.72) 3.5.2Temporalmotionofthebeam TheopticalattheendoftheM5beamlineismeanttofocusthebeamhorizon- tallyforitspassagethroughthebacklegholeinthering,whichhousesthe18mm 56mm (horizontalandverticaldimensions,respectively)superconductingmeanttocan- celsoutthesurroundingmagneticTheoutcomeofthisprocessisabeamwhose , ,and functionsdonotcorrespondtotheopticallatticefunctionsofthering.Also,the radialdispersionpriortothestorageringentranceisclosetozero;thus,storedmuonswill followtmomentum-dependentradialclosedorbitswithanoverallcoherencedueto theinjectionfavoringthestorageofspbetatronphases.Allthesefeaturesleadtoa mismatchedbeam. AsindicatedbyEq.( 3.58 ),thenonzeroaveragedbetatronamplitudesgiveplaceto oscillationsofthebeamcentroidsandwidthsovertime.Figures 3.47 and 3.48 depictsuch temporalbehaviorasseenatoneazimuthoftheringfromCOSY-basedsimulations.As explainedinSec. 3.4.1 ,nonlineartuneshiftsand(toalowerextent)therecombinationof high-andlow-momentummuonsyieldthede-coherenceofthetemporalmodulations,which areexpectedtohaveamplitudesoflessthanamillimeterbytheendofeach( ˘ 700 s). Thebeam-ringmismatchingislesssevereintheverticalphasespacethanintheradialcase thankstothenegligibleverticaldispersionandmorecomparablebeam-ringopticalfunctions. 133 Figure3.47:Long-termradialbeamde-coherence. FastFouriertransforms(FFT),showninFigs. 3.49 and 3.50 ,revealthetypicaltransverse frequenciesofthestoredbeam.Sinceinthisstroboscopicperspectivethebeamisobserved atonelocation,itsfrequenciesarecombinationsofcyclotronandradial/verticalfrequencies, listedinTable 3.7 . Table3.7:Transversemotionfrequenciesofthe g -2storedbeam(HV=18 : 3kV).Thelast columnindicatescyclotronrevolutionsperfrequencycycle. Frequency( f ) Expression[MHz][rad/ s]Period[ s]Cyclotronrevs. f c p 0 = (2 ˇm 0 ˆ 0 )6.705042.1280.149141.0000 f x f c x 6.332639.7890.157911.0588 f CBO ;x f c (1 x )0.372382.33972.685518.006 f y f c y 2.2181013.9370.450843.0228 f CBO ;y f c 1 y 4.486928.1920.222871.4943 InSec. 3.3 ,theconceptsofclosedorbitsandopticallatticefunctionswereelaboratedin thecontextofthe g -2storagering.Inthisway,adescriptionofamatchedbeamisexpected tobeexactinthelinearregime.Atthispointwherenonlinearitiesandbeammismatch aretlyquanthebeam-to-ringrelationsthatemergeinthisframework(i.e., 134 Figure3.48:Long-termverticalbeamtemporalmodulations. Eqs.( 3.53 )and( 3.57 ))aretested. Inthetemporaldimension,beamcentroidsandwidthsatazimuthallocationsaretracked fromsymplecticandnonlinearsimulations,andtheclosedorbitsandopticallatticefunctions arecalculatedwiththeCOSY-basedmodel.Figures 3.51 and 3.52 displayresultsat157 downstreamfromtheentranceofESQstationQ1S.AnelectricatESQstationQ1L withthetimedependenceexpectedduringRun-1a(seeSec. 3.6.1 )isalsoaccountedfor inthesetracking-vs-latticecomparisonsinordertotesttheframeworkbeyonditsnominal conditions.RandomtimeofthesizeofoscillationperiodsshowninTable 3.7 are appliedtotemporalcoordinatesfromsimulationresultsinordertowashoutfrequencies frombeammismatchinganddirectlycomparebeamdriftsoverthewholedatatakingperiod. Suchdriftsaretherelevantquantitytopayattentiontosincetheycanpotentiallybiasthe ! a measurement;ingeneral,ctsfromoscillationsdrivenbybeammismatchcancelout. BeamcentroidsdofollowclosedorbitsasindicatedinEq.( 3.57 )andshowninFig. 3.51 ; thebeamcentroidoscillateswiththeclosedorbitastheequilibriumpoint.Beam 135 Figure3.49:BeamfrequenciesextractedwithFFTfromcentroidsmotion. Figure3.50:BeamfrequenciesextractedwithFFTfromwidthsmotion. widths,ontheotherhand,requirealessstraightforwardextractionfromthelatticeside.In theradialcase,themomentumspread ˙ isrequiredtocalculatethedispersiveonthe radialwidth.Inaddition,emittancesarealsonecessarytobeknown.Intheexperiment, emittancesandthemomentumbeamdistributionsareobtainedexternallyfromFRanalysis [ 51 ]andstrawtrackerdata;intrackingsimulations,thesequantitiesaredirectlyobtained. Figure 3.52 showshowtheverticalwidthiswelldescribedbyEq.( 3.53 )whenthetime- evolvingverticalemittanceistakenintoaccount(seeFig. 3.53 ).However,thedescription oftheradialwidthinthetemporaldimensionfromEq.( 3.53 )isbyascalefactorof 136 Figure3.51:Comparisonbetweenclosedorbitandbeamcentroidfromtrackingat157 downstreamfromtheentranceofQ1S.Trackingdataisrandomizedtoremovebeambeating frommismatch.Theinitialdistributionandguidearepreparedforthe60 h caseas explainedinSec. 3.6 . Figure3.52:Comparisonbetweenbeamwidthfromopticallatticefunctions(greenandred entries)andbeamwidthfromtrackingat157 downstreamfromtheentranceofQ1S.The arediscussedinthemaintext.Trackingdataisrandomizedtoremovebeam beatingfrommismatch.TheinitialdistributionandguidearepreparedfortheRun- 1acaseasexplainedinSec. 3.6 .Theredlinecorrespondstoacasewithconstantemittances, whichintheverticalcaseplaysaroleindescribingaccuratelyverticalbeamwidths. 137 Figure3.53:Beamemittancesfromtracking.Duetothesmallerverticaladmittance,the verticalemittanceismorebymuonlossrates. about1 : 07.Thisdiscrepancyoriginatesfromtheskewnessoftheradial(andmismatched) distribution,whichdeviatesfromthepostulateofhavingellipticaldistributionsinphase space(or,equivalently,normaldistributionsperdimension).Nevertheless,thescalefactoris constantandthetimeevolutioniscapturedbyEq.( 3.53 ).Figures 3.54 and 3.55 showfrom thesimulationsidethephasespaceinbothtransversedirections,aswellastheirspatial projections(noteskewedradialdistribution). 3.5.3Beamazimuthalmodulation Theprecisereconstructionofmuoncoordinatesfromthestrawtrackingdetectormeasure- mentsprovidesreliabletime-dependenttransversemuonbeamintensityOnthe otherhand,thetwotrackingdetectorstationsarelimitedtotheirnarrowsensitivityof about5 , ˘ 1mupstreamfromtheirazimuthallocations(seeFig. 1.11 ).Allthingsconsid- ered,the24calorimetersaroundtheinnersideoftheringcouldideallybeusedtomeasure themuonbeamalongtheazimuth.However,thesedetectorswerespdesignedto 138 Figure3.54:Radialphasespaceatlatetimesofthedatatakingperiod(left)anditsspatial projection(right)fromRun-1simulation.Thepatterncloselyresemblesobservationsand itscharacteristicskewnessispresentinallRun-1datasets. measurepositronsenergyandarrivaltime(seeSec. 1.2.2 );atrustworthyreconstructionof themuonbeamtransverseoutofdetectedpositronsthathitcalorimetercrystals isthereforeduetoacceptance,resolution,andmaterialaspositronstravel throughtheringinstrumentation. Forthispurpose,theCOSY-basedmodelprovidesopticallatticefunctions|vwith reliabletrackingsimulations|inordertoextrapolatestrawtrackertransversebeamdata aroundthering.Theresultingextrapolationisusedtoquantifysystematicerrorsfromthe pitch,electric,andphaseacceptancecorrectionsto ! a .Itisalsoutilizedtoaveragethe magneticwiththemuonbeamaroundtheringforthecalculationof ~ B inEq.( 1.30 ). Usingopticallatticefunctionsismoreadvantageousthanbeamtrackingsimulationssince itisnotsubjecttofrequenciesfrombeammismatchnorfeaturesofthetrackingsimulation (e.g.,symplecticityandcomputationaltime).Oncedetailedmodelingoftheguide (undernormalorunexpectedscenariosasduringRun-1)inthestorageringisimplemented, opticallatticefunctionsarereproducedonacase-by-casebasis.Furthermore,ithasbeen 139 Figure3.55:Verticalphasespaceatlatetimesofthedatatakingperiod(left)anditsspatial projection(right)fromRun-1simulation. shownthatapropertransformationofthemeasuredbeamwithallitsfeaturesisalsopossible withtheopticallatticefunctionsforthepurposesoftheexperimentanalysis. BasedontestswiththeCOSY-basedmodel,theazimuthalbehaviorofthebeamisindeed reproducedbytheopticallatticefunctions.Sincetrackerdataprovidesinformationattwo azimuths,aneradialemittanceiscalculatedviaEq.( 3.53 ),where ˙ isextracted fromfastrotation(FR)analysis[ 51 ],themeasuredradialwidthistaken,andtheradial betaanddispersionfunctionsarecalculatedattrackerlocations.Intheverticalcase,the emittanceissimplycalculatedwithverticalwidthmeasurementsandverticalbetafunctions fromthemodel.Itisnecessarytowashoutthebeamtemporaloscillationswithdata randomization[ 108 ]oraveragingmeasurementswithintimerangesbeingmultiplestoCBO frequencies.Figures 3.56 and 3.57 showthegoodperformanceofopticalfunctionsforthe extractionofthebeamazimuthalbehavior.Eventhoughtheradialdistributionisnotwell behaved,thelatticefunctionscanbesafelyusedwithinerrorsof0 : 04mm. Toextrapolatethebeamtransverseintensity M ( x;y;t ; 1 ) ! M ( x;y;t ; 2 )muonby 140 Figure3.56:Comparisonbetweenclosedorbits(redcurves)andrandomizedbeamcentroids fromRun-1trackingsimulations. Figure3.57:Comparisonbetweenbeamwidthsfromopticallatticefunctions(redcurves) andrandomizedbeamwidthsfromRun-1trackingsimulations. muon,thefollowingexpressionisused: x ( 2 )= x ( 1 ; 2 )( x ( 1 ) x ( 1 ))+ x 0 ( 2 )+ D x ( 2 ) y ( 2 )= y ( 1 ; 2 )( y ( 1 ) y ( 1 ))+ y 0 ( 2 ) ; (3.73) where x ( 1 ; 2 )= p " x x ( 2 )+ D 2 x ( 2 ) ˙ ˙ x ( 1 ) and y ( 1 ; 2 )= s y ( 2 ) y ( 1 ) : (3.74) Inthisform,thetemporaloscillationsasmeasuredarecaptured.Analysisforthephase 141 acceptancecorrectionhasprovedthevalidityofthesetransformationswithtrackerdataas input[ 109 ].Forthemuonweightingofthemagneticsincethetemporalmodulationsare notrelevant,theentirebeamdistributionintegratedovertimeisshiftedandscaledaccording toEq.( 3.74 );seeSec. 4.5 formoredetails.Pitch[ 104 ]andcorrections[ 105 ]require amoredirectuseoftheCOSY-basedopticallatticefunctions. 3.5.4Time-momentumcorrelations AsexplainedinSec. 1.3.3 ,amomentum-timecorrelationafterbeaminjectionintroduces thelargestsystematicerrorofthecorrection.Atthetimeofthisdissertation,no proceduretomeasurethiscorrelationintheexperimenthasbeenestablished.Withthe COSY-basedmodel,theinitialdistributionattheexit(seeSec. 3.2.5 )istransferred intothestoragering,whoselongitudinal(Fig. 3.14 )andtheinjectionkickerspattern (Fig. 3.10 )determinetheinitialmomentum-timecorrelation.4 safterbeaminjection,the resultingcollimatedbeamisexpectedtoyieldamomentum-timedistributionasshownin 3.58 underRun-1akickersettings( 200 ; 170 ; 185Gasmaximumstrengths).Injectionof high-momentummuonsaheadofthelongitudinalcenterofthebeamisfavored;suchmuons requirelessoutwardkicktobestored,andtheringingofthekickerpulsebtheprocess. Thesamepatternoccursforhigh-momentummuonsontheothersideofthebeam,although toalowerextent.Figure 3.59 revealsthecharacteristicshapeofthebeammomentum spreadfromFRanalysis,whichindicatesaproperrecreationoftheinjectionprocessin theCOSY-basedsimulation.Other g -2collaboratorsproducesimilarsimulationsofthe momentum-timedistribution[ 110 ]andtheoutputisqualitativelyequivalent,althoughthe COSY-basedsimulationexhibitsastrongercorrelation;collaborativeareinprocessto thecalculationofthesystematicerrorassociatedwiththemomentum-timecorrelation 142 Figure3.58:TypicalRun-1momentumtimebeamdistributionfromRun-1trackingsimu- lation. Figure3.59:ProjectionsofthetypicalRun-1momentumtimebeamdistributionfromRun-1 trackingsimulation. 143 for g -2runsposteriortoRun-1. 3.6BeamDynamicsofRun-1 DuringallthefourRun-1datasets(inchronologicalorder 60h (1a), HK (1b), 9d (1c),and EG (1d)),measurementsofthetransversemuonbeamfromthestrawtrackerdetectors revealedunexpecteddriftsofthebeamcentroidandwidthoverafractionofthe[ 86 ] (i.e., t . 200 safterbeaminjection).Furthermore,CBOfrequenciesoftheradialcentroid oscillationswerealsofoundtoevolveduringthedatatakingperiod,slowlyconvergingto theirnominalvaluesoverthecourseofa UndernominalconditionstheESQstabilizesafter t ˇ 30 s,inwhichcasetheguide areconstantandthustheopticallattice,too.Inthisnormalscenario,CBOfrequenciesdo notchangesincethestablelatticeprovidesconstantbetatrontunes(seeTable 3.7 ).Also, closedorbitsareexpectedtobestableand,consequently,thepointsaroundwhich beamcentroidsoscillateareindeedAspresentedinSec. 3.5.2 ,transverseemittances aresensitivetomuonlossratesasexperiencedduringRun-1;therefore,beamwidthscould driftunderastableopticallattice.Alltheseexpectationsservedtointerpretobservations anddeterminethespbehavioroftheopticallatticeduringRun-1. Inparticular,CBOfrequenciesrelatetogradients,andcentroidcoordinatesresult fromdipolesteering.Lessambiguouslyandintheviewof2Dmultipolesexpan- sion,normalquadrupoletermsstronglyrelatetotheradialCBOfrequencyviaradialtunes andskewdipoletermslongtermmotionoftheverticalbeamcentroidviavertical closedorbits.Beingmindfulofsuchlattice-beamrelations,measurementsoftheradialCBO frequencyandverticalbeamcentroidoverthefourRun-1datasets(seeFigs. 3.60 and 3.61 ) 144 aretakenasinputforthereconstructionoftheunmeasuredRun-1ESQelectricds. Figure3.60:RadialCBOfrequenciesduringthefourRun-1datasetsfromstrawtracking detectorsdata.Semi-transparentmarkersareobtainedfromslidingsinusoidal( 5 s) totherecordedradialbeamcentroid,whereassolidlinesresultfrommulti-parameter throughtheentireBothmethodsareequivalent. Ontheotherhand,ahandfulofpottedresistors(whichconnecttoESQplatesaspartof theRCcircuitry)withHVRsinserieswereinstalledduringRun-1forlogisticalreasons[ 111 ] (seeFig. 3.62 ).Twoofthem,connectedtothetopandbottomlongplatesatESQstation Q1(\Q1LT"and\Q1LB"),weremeasuredforseveralhighvoltageswithaHVprobeafter Run-1concludedandfoundtobedamaged.TheirHVtraceswereexpectedtobehavelikethe nominalonesshownwiththinlinesinFig. 3.63 .However,measurementsdisplayedlonger relaxationtimes(circularmarkersinFig. 3.63 ).Moreover,thepeculiarHVtracesfrom thedamagedresistorsmanifestedoverthecourseoftheprobemeasurements, providingstrongevidenceofmultimodalstagesoftheESQstationQ1LduringRun-1(also fromtrackerdatashowninFigs. 3.60 and 3.61 ).Thefaultybehaviorofthesetwo resistors(outofallthethirty-twoinstalledintheESQcircuitry)indicatedcoronadischarges ontheresistorsurface[ 112 ];resistorsoutgassedwhiletheirtemperatureraised,leadingto 145 Figure3.61:VerticalcentroiddriftsduringRun-1fromstrawtrackingdetectorsdata.Solid linesarewithdoubleexponentialtermsandaconstantpartasthefunctionalform. thedischargeatlowvoltages.Theoutgassingwaslikelymoredramaticduringthelastand longestRun-1dataset(1d),pushingtheESQtoproducemoreunusualelectricguide InSec. 3.6.1 ,amethodtoreverse-engineertheQ1LT-andQ1LB-HVtracesperRun-1 datasetiselaborated.Fromthereconstructionresults,derivedopticallatticefunctionsare presentedinSec. 3.6.2 . Section 3.6.3 describesareliablemethodtoreconstructthebeam6Dstructureoutof experimentaldata.Lastly,inSec. 3.6.4 thestoredmuonbeamfromtrackingsimulationsis presented,withthereconstructedHVtracesanddata-basedinitialbeamconditionsasinput fortheCOSY-basedmodel. 146 Figure3.62:Single-CADDOCKhigh-voltageresistor(top)andchainofpottedHVresistors (bottom).TwoofthelattertypeofresistorsbecamedamagedduringRun-1. 3.6.1ReconstructionoftheelectricguideduringRun-1 Technique TheelectrostaticpotentialproducedatanESQstationcanberepresentedasasuperposition ofthefourcontributionsoriginatedbyeachofitstop\T",bottom\B",inner\I",andouter \O"plates.Inadditiontothepotential V 0 ( x;y;t )expectedatESQ-stationQ1Lunder nominalconditions,anadditionalcontribution V T ( x;y;t )and V B ( x;y;t )fromthefaulty Q1LTandQ1LBplates,respectively,duringRun-1isoverlaidasaperturbationunderthe straight-platesapproximationasfollows: V ( x;y;t )= V T ( x;y;t )+ V B ( x;y;t ) = 1 X k =0 1 X l =0 HV T ( t ) g k;l + HV B ( t ) b k;l x k y l (3.75) InEq.( 3.75 ), HV T ( t )and HV B ( t )aretheextrahighvoltages\HV"onthetopand bottomplatesduetothedamagedresistorsinRun-1suchthatthetotalHVtracesaregiven 147 Figure3.63:HV-tracessample(circlemarkers)fromHVprobemeasurementsinSeptember 2018,atQ1Lplatesconnectedtothedamagedresistors.Blueandredlinesdepictnominal HVtraces. by HV T ( t )= HV 0 ;T ( t )+ HV T ( t )and HV B ( t )= HV 0 ;B ( t )+ HV B ( t ) ; (3.76) where HV 0 isthenominalcase.Thecots g k;l and b k;l determinethedistribution of HV T;B ( t )amongthetop/bottomplatemultipoles.Since b 1 ; 0 =0, b 1 ; 1 =0,and b k;l =( 1) k + l g k;l duetotheorientationand180 rotationalsymmetryofthetop/bottom plates,Eq.( 3.75 )canberewrittenas V ( x;y;t )= V s dip ( t ) y + V n quad ( t )( x 2 y 2 )+ ; (3.77) 148 where V s dip ( t )= HV T ( t ) HV B ( t )] b 0 ; 1 and V n quad ( t )= HV T ( t )+ HV B ( t )] b 2 ; 0 : (3.78) Thestudy[ 89 ]providesthecots b 0 ; 1 and b 2 ; 0 .Giventheorthogonalityofthetwo equationsinEq.( 3.78 )intermsof HV T and HV B ,thereisauniquesetoftopand bottomHVtracesthatyield V s dip ( t )and V n quad ( t ).Toevaluatethesetraces,theextra skewdipole V s dip ( t )andnormalquadrupole V n quad ( t )termsmustbelinkedtobeam dynamicobservablesmeasuredbythe g -2strawtrackingdetectors,asshownnext. Implementation Underthepresenceoftheextraverticaldipoleelectricpotential V s dip ( t )(Eqs.( 3.77 )and ( 3.78 )),theverticalclosedorbitbecomesdistorted.Therefore,bymeasuringthedistortion oftheverticalclosedorbitatoneazimuthallocationofthestoragering(akavertical points\ y 0 ")overtime, V s dip ( t )canbequanInfact,thestrawtrackershavethe abilitytoextractsuchverticalbeamequilibriumpositionsaroundspclocationswithin thering. Toillustratetherelationbetween V s dip ( t )andtheobservable y 0 (equivalenttothenon- oscillatingverticalmeanfromtrackerdata),theformervariablecanbetreatedasadipole steeringerror[ 113 ]: 0 B @ y b 1 C A 0 = I M y 0 1 0 B @ 0 y 1 C A ; y ˇ e V s dip E 0 l r ref (3.79) where M y 0 istheverticalquadrantofthestorageringtransfermapwithoutthesteering 149 error, y istheresultingverticalsteeringangle, E 0 theenergy, r ref thereferenceradius of V s dip ,and l isthelengthoftheelementthatprovidesthesteeringerror.Itisworth mentioningthatthelinearverticaltransfermap M y 0 hastoaccountforthegradienterror explainednext. Thenormalquadrupoleextraterm V n quad ( t )introducesadistortiontotheradialdefo- cusinggradientatQ1L.Consequently,thebetatrontunes x;y andbeamtransversalwidths areduetothenonzero V n quad ( t ).TrackerscanindirectlymeasuretheradialCBO frequency(seeFig. 3.60 ), ! CBO ,whichrelatestothetunesthroughthecyclotronfrequency, f C ,via ! CBO ˇ 2 ˇf C (1 x ).Inasimilarfashion,therelationbetween V n quad ( t )and theobservable ! CBO canbeelucidatedbytreatingtheactionof V n quad ( t )asagradient error[ 78 ]: x =1 ! CBO 2 ˇf C = 1 2 ˇ 4cos 1 Tr( M x 0 ) 2 + e V n quad pv x l r 2 ref ! ; (3.80) whereTr( M x 0 )isthetraceofthehorizontalquadrantofthestorageringtransfermapwithout thegradienterror.Inreality,theactionoftheextraskewdipoleandnormalquadrupole termsatQ1LduringRun-1isentangledandmagneticinhomogeneitiesalreadydistort closedorbits.Moreover,trackersdonotmeasuretheverticalclosedorbitatQ1L.Thus,the illustrativebutsimplisticequations( 3.79 )and( 3.80 )donottosolvefor HV T;B ( t )via V n quad ( t ) ; V s dip ( t ) with( y 0 ;! CBO )fromtrackerdata.Forthispurpose,withthehigh- yCOSY-basedstorageringmodelandoptimizationalgorithmssupportedbyCOSY 150 INFINITY[ 53 , 54 ],amorerepresentativesetofbijectiveequationsisprepared: y 0 ( t )= F 1 HV T HV B ; ~ A ; t ) ; (3.81) ! CBO ( t )= F 2 HV T + HV B ; ~ A ; t ) : (3.82) Withtheserelationsfullyestablished,theQ1LTandQ1LBhigh-voltagetracesarerecon- structed.Thevector ~ A containsalltheothernominalparametersofthestorageringin- dependentofthebadresistorsbehavior.Fig. 3.64 illustratestheoptimizationprocessfor onesetoftrackermeasurements( y 0 ;! CBO )toobtaintheoptimaltopandbottomHV-trace valuesatasptime.InFig. 3.64 ,rightplot,\ f obj "isthesumofobjectivefunctions thattheoptimizerminimizesiteratively: f obj; 1 = 1 sim x HV T ; HV B ) Trackerdata x ! and f obj; 2 = 1 y sim 0 HV T ; HV B ) y Trackerdata 0 ! ; (3.83) wherethesuperscript\sim"standsforthevaluesfromtheCOSY-basedmodel,dependent onthe HV T and HV B valuesinputtothemodel. ReconstructionResults Fig. 3.65 presentstheHVtracesfromQ1LTandQ1LBduringRun-1datasetsasrecon- structedwiththemethoddescribedabove.Forthetimewindowpriortothenominal measurementstarttime, t< 30 s,theHV-tracesreconstructionfailstooutputresultsthat resemblethefunctionalformsasdirectlymeasuredwiththeprobe.Tobypasssuchlimi- tation,thefunctionalformofthereconstructedHVtracesat t> 30 sisextendedto outthegapat t< 30 s.TheimplementationofthereconstructedHVtracesisvalidated 151 Figure3.64:HighvoltageonQ1LTandQ1LBplates(left)andobjectivefunction(right) periterationastheoptimizationtakesplacefortheRun-1,EndGamedataset,at40 s. TheoptimizersupportedbyCOSYINFINITY(i.e.,thegeneralizedleastsquaresNewton method)Q1LT/Q1LBHVssuchthattheCOSY-basedstorageringmodelwithdamaged resistorsaccountedforreproducesaverticalpoint(equivalenttotheverticalmean) andCBOfrequencyasmeasuredbytrackerstation12. Figure3.65:ReconstructedHVtraces(normalizedtotheirnominalvoltageset-points)at Q1LduringRun-1datasets.DashedandsolidlinescorrespondtoQ1LTandQ1LBplates, respectively. 152 bycomparingbeamtrackingsimulationresultswithtrackerdata,i.e.,CBOfrequenciesand verticalcentroidsovertimeasshowninFig. 3.66 . Figure3.66:Comparisonsbetweentrackerdata(bluemarkers)andtrackingsimulations (redmarkers)attrackerS12azimuthalreadoutlocationwiththedamaged-resistors implemented.OntheleftistheCBOfrequency,whereasontherightverticalcentroidsare shown.Similarly,strongagreementsareobtainedfortheotherRun-1datasets.Forcompar- isonpurposes,theverticalentriesontherightplotareshiftedwithintrackermisalignment errors. Crosschecksanduncertaintyestimation TheHV-tracesreconstructionraisesfromtheobservationofslowlychangingbeamparame- ters(CBOfrequencyandverticalmean)attheazimuthallocationsintheringwheretrackers takedatafrom.TheCBOfrequencyisaglobalparameterindependentoftheringlocation, asitisdirectlyrelatedtothebetatrontunes,anditsdirectextractionfromradialbeam oscillationsandfunctionalhasdemonstratedtobereliable.Ontheotherhand,the early-to-lateverticalmeandriftismoresubjecttothelimitedstatisticsoftheavailabledata, especiallyatlatetimesintheTocompensateforthisaspect,verticalmeandriftsas recordedbycalorimetersaroundtheringserveasanextrainputtoconstraintheequilibrium verticalmeanaftertheofthedamagedresistorsduringthethiswayguaranteeing theoverallverticaldriftsfromtrackerdata. 153 Figure3.67:Blackmarkersareverticalbeamdrifts(40-300 s)fromcalorimeterdata(energy thresholdof1.7GeV)undermcorrections,wheredoubleexponentialwere employed.Inblue,verticalclosedorbitdistortiondriftsfrom40-300 susingtheCOSY- basedmodelwiththereconstructedHVtracesimplemented.Theblueerrorbandisan estimateoftheuncertaintyintroducedbytheverticaldispersion.Theredsquaresarefrom a gm2ringsim muontrackingsimulation[ 45 ]withanimplementationoftheelectricguide reconstructedwiththeCOSY-basedmodel.Thesimulationsarematchedtodataby associatingtheclosedorbitplaced ˘ 22 upstreamofthecalorimeterposition. Moreover,severalaspectsfromthestorageringmodelingthatcouldinterferewiththe sensitivityoftheverticalbeammeanandCBOfrequencytochangesintheguidewere considered[ 114 ],namely: MagneticmultipoleconterrorsfromTrolleydata(notgreaterthan300ppb). HVreconstructionbasedondatafromtrackerstationS12vs.S18. Ambiguityofmagneticreferenceazimuthalangle. ThelargestimpactofthelistedonthereconstructedHVtracescamefrommag- neticcoterrorsand,forinstance,theircorrespondingopticallatticefunctionsledto systematicerrorsofthephaseacceptancecorrectionofabout1ppb. ThepresenceofathirddamagedresistoratstationQ3Lwasalsosuspected.However, whenareconstructionofthreetraceswasattemptedundertheseconditions(information 154 oftheverticalbeamatthetwotrackerstationswasaccountedforsimultaneously,tocon- siderthreeoptimizationobjectivefunctions),theoptimizationmethoddidnotconvergeto physicalresults.Ataqualitativelevel,HVmeasurementsofthethirdpossibleplateto havebeendamagedduringRun-1exhibitedastablebehaviorafterthestarttime[ 111 ]. Also,comparisonsshowninFig. 3.67 consideringtwodamagedresistorsrecreatecalorimeter observationswithinexperimentalerrors. 3.6.2OpticallatticefunctionsduringRun-1 WiththereconstructedHVtracesatQ1LTandQ1LBforeachRun-1dataset,thetotal electricpotentialatQ1LisintheCOSY-basedmodelandopticallatticefunctions prepared(seeSec. 3.3.2 ). Figure 3.68 showshowverticalclosedorbitsareshiftedduetothefromdamaged resistors.TheminimumdistortionontheverticalorbitoccursatQ1L,wherethesteering errorislocated.Attheoppositesideofthering,theverticalclosedorbitsuitslargest drift.Asimilarpatternarisesfromtheverticalbetafunctionsbeating(seeFigs. 3.69 and 3.72 ).Theringisaclosedsystemwhereextrasteeringorgradienterrorsbeyonddesign haveglobalimplicationsontheentirelattice.Forinstance,whilemuonscrossthemis- poweredstationQ1L,thelowerfocusingverticalgradientletsthebeamspreadsouttoa largeramountthanthedesigncase,whereinverticalbetafunctionsaproportional azimuthalevolution. Ontheradialend,largerradialtuneschangetheresultingpatternsdrivenbyQ1Lduring Run-1(seeFigs. 3.70 and 3.71 ).Theradialclosedorbitisminimallybythedamaged resistors;thetopandbottomplatesatQ1Ldonotintroducenormaldipoleelectric AstheQ1LTandQ1LBplatesslowlyconvergetothenominalHVsetting,alloptical 155 Figure3.68:Verticalclosedorbitsat30 s(redcurves),100 s(bluecurves),200 s(green curves),and300 s(blackcurves)duringRun-1.GrayshadowsdepictESQstationsalong theazimuth,wheretheQ1Supstreamedgeisat =0.Orangelinesindicatecollimator locations.RedcurvesaresubjecttotheoftheESQscrapingandthe greencurveshavealmostreachedtheequilibriumvalues. 156 Figure3.69:Verticalbetafunctionsat30 s(redcurves),100 s(bluecurves),200 s(green curves),and300 s(blackcurves)duringRun-1. functionsadiabaticallyconvergetoastablestate.Figure 3.73 showsthetimeevolutionof thelatticeat210 fromtheupstreamentranceofstationQ1S(Run-1dataset1d),wherethe latticegradientsarethelargest. Withthesedataset-by-datasetperiodicfunctionsoftheopticallattice,trackerdatais extrapolatedaroundthering(seeSec. 3.5.3 ).Figure 3.74 displaysthebeamcentroidsand widthsbasedontrackerdata(EG),station12. 157 Figure3.70:Radialbetafunctionsat30 s(redcurves),100 s(bluecurves),200 s(green curves),and300 s(blackcurves)duringRun-1. 3.6.3Initialbeamdistributions Themuon g -2strawtrackingdetectorsystemhasthebeam-diagnosiscapabilityofrecording radial x andvertical y muoncoordinatestoresolutionsof ˙ ˇ 4mmandtime t coordinates with3.4nsprecisionduringRun-1[ 38 ]. Inthestroboscopicperspective,trackermeasurementscanbetreatedaslocalizedina wazimuthallocationasaresultofthenarrowazimuthalsensitivity( ˇ 4 : 9 )and theweaklyfocusingopticsapproximation.Inthisview,theradialmotionofthebeam ismodeledforsmalltimerangesasanensembleofmonochromaticmuonsoscillatingat acommonradialCBOfrequency ! CBO withtequilibriumpositions x i e ,betatron 158 Figure3.71:Radialdispersionfunctionsat30 s(redcurves),100 s(bluecurves),200 s (greencurves),and300 s(blackcurves)duringRun-1. amplitudes A j ,andphases ˚ k : x ( t )= x i e + A j cos( ! CBO t + ˚ k ) : (3.84) Themeasureddata-binintensity N mn atcoordinates( x m ;t n )|seeFig. 3.75 |canbeex- pressedas[ 115 ] N mn = n i X i =1 n j X j =1 n k X i = k ijkmn f ijk ; (3.85) where f ijk istherelativemuonpopulationidenbytheset f x i e ;A j ;˚ k g fromthebeam probabilitydensityfunction(PDF) f ( x e ;A;˚ ).InEq.( 3.85 ),thesummationupperbounds 159 Figure3.72:Relativeverticalbetafunctionsdriftfrom30 sto1000 s. correspondtothenumberofvaluessubjecttothebinningsizealongeachcoordinatein f ijk (e.g., n i =45for x 1 e =[ 45 ; 43]mm, x 2 e =[ 43 ; 41]mm, ::: , x 45 e =[43 ; 45]mm). Ontheotherhand,therelativenumberofmuonswithsp x i e ;A j ;˚ k coordinates thatcontributetothemeasuredintensity N mn ,symbolizedby ijkmn inEq.( 3.85 )andnot tobeconfusedwithbetatronfunctions,isequaltotheprobabilityofsuchmuonstobefound atthebin( x m ;t n ): ijkmn = Z r + x m = r Z t + t m = t ( x m ) p 2 ˇ˙ e 1 2 ˙ 2 h x m x i e + x + A j cos ( ! CBO t n + ˚ k ) 2 dx m dt n : (3.86) Theradialacceptance ( x )inEq.( 3.86 )[ 38 ]accountsfordetectionectsofthetracker stations.The x isdeterminedbasedondataandaccountsforclosedorbitdistortions, trackerreconstructionandtrackermisalignments. Withthesewell-establishedrelationsbetweenbeamPDFsand x t trackerdata,the 160 Figure3.73: x , y , D x ,andverticalclosedorbitat210 fromtheupstreamentranceof stationQ1SfordatasetRun-1d. optimal f ijk functionscanbeestimatedvianon-negativeleastsquares(NNLS)minimization ofthefollowing ˜ 2 expression: ˜ 2 = n m X m =1 n n X n =1 P i;j;k ijkmn f ijk N mn 2 ˙ 2 N mn : (3.87) Underconstraintsinspiredbyfurtherdataanalysis,theminimizationconvergestophysical solutionsafterlessthan10,000iterations.Forinstance,theFRsignal[ 51 ]enablestolimit theconditionalPDF f ( x e j A;˚ ),whereinresultingdeviationsfromtheoriginalsignalcanbe attributedtodibetweencalorimeterandtrackerdetection 161 Figure3.74:Beamdriftsfrom40 sto300 sanchoredtotrackerdatafortheRun-1d dataset.GrayshadowsdepictESQstationsalongtheazimuth,wheretheQ1Supstream edgeisat =0.Orangelinesindicatecollimatorlocations.WiththeRun-1opticallattice functions,theobservedbeamdriftsmeasuredbythemuon g -2strawtrackingdetectorsare projectedalongtheentireazimuthofthestoragering. Figure 3.76 showsatypical f ( x e ;A;˚ )functionprojectedalongtwodimensions.Correla- tionsbetween x e , A ,and ˚ areexpectedtoresultfromthedynamicscausedbytheinjection, collimation,andcirculationofthemuonbeaminthestoragering[ 116 ]. Thankstotheperiodicityofthemismatchedbeammotionovertime,adatarangeof oneCBOperiodisttoreconstructthebeamconditions(andnecessarytoignore betatrontuneshifts).Inordertocapturethebeamscrapingprocess(anessentialcomponent forrealisticmuonlossratesmodeling)thattookplaceduringRun-1thePDFsarecalculated fromdataataround4 safterinjectionwhilethescrapingschemeisatitsinitialstage. Fromthevertical-motionfront,thereconstructionfollowsthesameprocessasforthe radialcasewithasimofallverticaloscillationsfollowingthesamepointin 162 Figure3.75:Resolution-andacceptance-correctedradialpositionversustimefromtracker data,Station12.Byreconstructingthetrajectoryofdecaypositronsdetectedbythetracking planes,muondecaypositionsareextrapolated.Coherentoscillationsareaconsequenceof thebeaminjectionprocess. theabsenceofverticaldispersion.Sp,theverticaldistributionofthemuonbeamis describedwithverticalbetatronfunctionsandconstantoscillation Theinitialtimeofthebeamissetbasedonmeasurementsfromscintillatingde- tectors.WiththeCOSY-basedmodel,whichincludestheobservedinjection-kickerringing patternduringRun-1andarealisticbeamdistributionatinjection(seeSec. 3.2.5 ),acor- relationbetweentimeandmomentumisassignedtotheinitialbeamdistributionat 4 s. Usingthelinearbeamdynamicsframework,theinitial ~z i raycoordinatesetsareprepared foreachmuonfromthereconstructedPDFsfornonlinearbeamtrackingsimulationswith theCOSY-basedmodel. Eventhoughnonlinearitiesarenotconsideredinthereconstructionoftheinitialbeam 163 Figure3.76:Reconstructedbeamdistributionintheradialdirection.Ontheleftplot, theupperlimitsaredeterminedbythephysicalaperturesofthecollimatorswhichbound maximumradialexcursions,whereasthemaximumkickerstrengththelowerlimits. conditions,allthede-coherence,betatronamplitudemodulation,andtune-dependentshifts originatedbythehigherorderguidearemanifestedinthenonlinearbeamtracking simulations. Thisreconstructedmethodassumesnocorrelationsbetweentheradialandverticalmo- tion,whichagreeswithobservationsfromthestrawtrackingdetectors.tlylarge skewquadrupolecotsfromtheelectricandmagneticcanpotentiallyintroduce x - y beamcorrelations,andpropermodelingoftheseinthesimulationgivesriseto such 3.6.4ThesimulatedRun-1beam Thefollowingfeaturesdevelopedinthisdissertationsetthestageforahighlyrealistic recreationofthestoredmuonbeamduringRun-1: COSY-basedmodelofthemuon g -2storagering. ReconstructedESQatstationQ1Lbasedonexperimentaldata. Realisticinitialbeamdistributionbasedonexperimentaldata. 164 Figure3.77:Muonbeamintensityfromsimulation(left)andtrackerdata(right),station 12,duringRun-1afrom30 s 2).Figure 4.20 showsresultsforthisscenario.Undertheseapproximations, j C e j ˝ ! a ! a 0 ˛ sim =0 : 8 0 : 1ppb : (4.27) Thesmallislikelycausedbythecurvatureoftheplates,sincethesimulation accountsuptotenth-ordertermsinthetransfermaps.Whenthemomentumspreaddistri- butionfromFRanalysisisusedinstead: j C e j ˝ ! a ! a 0 ˛ sim =5 : 2 0 : 1ppb : (4.28) Thus,theisolatedofanasymmetricRun-1style distributionaddsacorrectionof ˘ 4 : 4ppbto C e ,whichintroduceshigherorderdistributionmoments(e.g., h 3 i )tothe 201 Figure4.20: h ! a i spreaddistribution(right)andversustime(left).Muonsarecontained withinthehorizontalmidplanetoexclude C p .Typicalfrequencyspreadsduetothe C e correctionareapproximatelyequalto ˘ 540ppb,asobservedfromthestandarddeviation intherightplot. momentumspreadsuchthat ˝ ! a ! a 0 ˛ = ˝ e m 1 1 1 (1+ ) 2 z E x c = j C e j + n 0 2 0 1 n 0 1+2 2 0 D 3 E + 6 = j C e j : (4.29) Inasimilarmanner,furtherfeaturesofthestoragering|notaccountedforinthesensible approximationsfromwhich C e and C p arederived|areaddedinthetracking,namely,ESQ platesdiscretization,ESQhigherordermultipoletermsforall a k; 0 cots,andESQ fringe(seeSec. 3.2.1 ).Table 4.4 listsallthecasesofstudy,inadditiontothebeam andringfeaturesdescribedabove(i.e.,\DIEQRing"),andTable 4.5 showscomparisons between C e andtrackingforeachofthosecases. 202 Table4.4:Casesofstudyfor h ! a i tracking,fromthesimple(top)tothedetailedmodel (bottom). Caselabel ESQPlatesdiscretization ESQhigherordermultipoles ESQFringe DIEQRing DIEQ X DIEM X X DIEM-FR X X X Table4.5: C e versustracking(noverticalbetatronmotion). Case j C e j D ! a ! a 0 E sim [ppb] DIEQRing 4 : 4 0 : 1 DIEQ 5 : 5 0 : 1 DIEM 9 : 8 0 : 1 DIEM-FR 7 : 7 0 : 1 Whiletrackingresultsindicateanoverall ! a frequencyslightlyhigherthan C e duetoan asymmetricmomentumspread,theotherpushesthediscrepancydownupto7 : 7ppb. Thedisagreementsareinducedbymuonswithmomentumnearthemomentumac- ceptanceofthering,asindicatedby -binnedtiatedresults. 4.4.3Pitchcorrection C p Uptothispoint,noverticalmotionofthebeamhasbeenconsideredinordertoanalyze C e only,withouttheintrusionof C p .Next,theattentionisdirectedto C p instead.For thispurpose,amonochromaticbeamcomposedofmuonsatmagicmomentumonly(i.e., =0)istrackeddown.Togetclosertotheinitialideasthatpermittedthederivation C p , muonsinitiallyhavenoradialbetatronmotioneventhoughanunavoidable,andsmall,radial motionemergesfromthecurvatureoftheESQplates( E x ( x =0 ;y 6 =0) 6 =0).Nevertheless, sinceradialandverticalmotionsaremostlydecoupled,thesmallradialmotionsdonot 203 incorporateerrorstothe C p versustrackingcomparisonsdiscussednext. InthisscenarioandkeepingperfectverticalmagneticaswellascontinuousESQ plateswithastructureoriginatedonlybythe a 2 ; 0 quadrupolecot(seeSec. 3.2.1 ), resultsasshowninFig. 4.21 areobtained.Incontrasttothe C e -onlystudiesabovewhere Figure4.21: h ! a i spreaddistribution(right)andversustime(left).Muonsarelaunched withnoradialmotionnormomentumtoexclude C e .Thefrequenciesspreaddueto the C p correctionis0 : 1%asshownintheplotontheright. individual ! a 'sarespreadoutwithinastandarddeviationof ˘ 537ppb,the C p -onlycase exhibitstypicalspreadsofabout0 : 1%.Assuch,theassociatedstatisticalerrorislarger with N muons =128 10 6 and N turns =100;namely,8 : 5ppb.Thecomparisonyields C p ˝ ! a ! a 0 ˛ sim = 9 : 1 8 : 5ppb : (4.30) Oneofthemainassumptionsof C p isharmonicverticalmotion( y ( t )= y 0 cos ! y t + ˚ y ). However,tosatisfyLaplace'sequationincurvilinearcoordinatestheverticalelectric thatdrivestheoscillatoryverticalmotionisnotpurelylinear.Instead,nonlinearcomponents 204 themotionaswell: E y ( x =0 ;y )= 1 X l =1 a 0 ;l y l 1 ( l 1)! : (4.31) Table 3.1 showsnonzero a 0 ;l 'sfor l =0 ; 2 ; 4 ;::: duetotheESQstationsgeometry.Since thetrackingsimulationsencompasssuchcomponentsoftheESQ C p isnotinfull agreementwiththetrackingresults(Eq.( 4.30 ))forwhichtheverticalmotionisnotentirely harmonic. 4.4.4Electricandpitchcorrections C e + C p Inreality,themeasuredprecessionfrequencyofthe g -2storedbeamexperiencesbiasing duetoboththatthe C e and C p attempttocorrect.Uptonow,thecomparisons betweentrackingsimulationresultsandthesecorrectionstreatedindependentlyhaveyielded reassuranceontheirusageto ˘ 10ppbaccuracy.Thenextstepistotrackthefullbeam withbetatronamplitudesinbothtransversedirectionsandrealisticmomentumspread,as spatthebeginningofthissection.Thespreadof ! a isdominatedbythepitch asshowninFig. 4.22 .Table 4.6 listscomparisonresultsfortheimplementedcases. Table4.6: C e + C p versustracking(fullbetatronmotion). Case C e + C p D ! a ! a 0 E sim [ppb] DIEQRing 6 : 4 8 : 5 DIEQ 8 : 7 8 : 5 DIEM-FR 10 : 0 8 : 5 Duetothecircularstoragevolume,muonswithsmallerverticalbetatronamplitudes y max 205 Figure4.22: h ! a i spreaddistributionversustimeofasimulated g -2beamwithRun-1 characteristics(notethelogarithmiccolorscale).Thepitchedominatesthespread. areallowedtohavemomentumwithintheentiremomentumacceptanceofthering. Ontheotherhand,thelargest y max amplitudesareallowedonlyfor ! 0,inwhichcasethe correctionisminimal.ThisinterrelationisrevealedinFig. 4.23 (leftplot);muonsnear themidplanearemoreinneedofancorrection.Thebeamisnormallydistributed inphasespace;therefore,thepopulationofmuonswithsmall y max amplitudestendsto besmallandresultsinFig. 4.23 areweightedaccordinglytocomputethetotalprecession frequencyofthebeam. 4.4.5Run-1considerations DuringRun-1,boththeindex(Fig. 4.24 ),inEq.( 3.26 ),andradialdispersion function(Fig. 3.71 )driftedduringthedatatakingperiod.Asaconsequence,theensemble ofmomentum-dependentradialclosedorbits x e = x 0 + D x slowlyvariedovertimeaswell. 206 Figure4.23: h ! a i spreadbinnedoververticalbetatronamplitudes(left)andmomentum (right)forthecaseoffullbetatronmotion. Figure4.24:iveindicesovertimeduringRun-1fromstrawtrackersdata( n 0 = 1 1 ! CBO;x = 2 ˇf c 2 ). 207 Moreover,trackingsimulationswiththeCOSY-basedmodelindicateaslightlychanging momentumspreaddistributionduetolostmuons (30 s ! 300 s) = (30 s) ˇ 0 : 25% and RMS (30 s ! 300 s) RMS (30 s) ˇ 0 : 4%).Bytakingoneoftheintermediate expressionsinvolvedinthederivationofthestandarddcorrection,seeEq.( 4.22 ),such canbeanalyzedasfollows[ 129 ]: C e ( t )=2 n 0 ( t ) 2 0 ˆ 0 h x e ( t ) ( t ) i N;' ) C e ( t )=2 n 0 ( t ) 2 0 ˆ 0 h h D x ( t ) i ' D ( t ) 2 E N + h x 0 ( t ) i ' h ( t ) i N i ; (4.32) where hi N denotesaverageoverthemuonbunchand hi ' areazimuthalaveragesalongthe ringwithESQoccupancy.Themomentum-independentclosedorbit x 0 wasnottly bythechangingelectricelds,thusitscontributiontoachanging C e isnegligible. Figure 4.25 showshowthetemporary C e changesduetothedynamicofRun-1, relativetothenormalcalculationofthecorrection C e 0 C e = C e 0 C e ( t )).Toestimate theentireovereachRun-1datasetduetothetime-evolving ! a fromchangingelectric andmomentumdistribution,resultsshowninFig. 4.25 areincorporatedintoMonte CarlosimulationssimilartothemethodfollowedinSec. 4.2 toextract C ml but,instead ofamanipulationof ' 0 ,thespinprecessionfrequency ! a isdirectlyas ! a ( t )= ! a 0 (1 C e ( t )).Table 4.7 listsdeviationsfromtime-independent C e corrections;allofwhich arefoundtobesmallcomparedtothemainsystematiccorrectionsof C e duringRun-1[ 45 ]. Theectofchangingverticalbeamwidthson C p hasalsobeenstudiedwith y fromthe COSY-basedmodelfunctions[ 104 ]andfoundtobesmall( ˘ 1ppb). Inconclusion,thisanalysishastestedtherobustnessof C e and C p andindicatesthat theasymmetricmomentumspread,discreteESQplates,nonlinearESQanditsfringe introduceasystematiccorrectionof 10 : 0 8 : 5ppbtothestandard C e + C p formalism 208 Figure4.25:Temporal C e ( t )corrections(relativetothetime-independentcase)duringRun-1 underthesofchangingmomentumspreadandtime-dependentESQelectric usedintheexperiment. 4.5TheWeightedMagneticField ~ B Thedriverofthemuonbeamrotationalpolarizationarounditsmomentumdirectionisthe magneticofthe g -2storagering.Subjecttothespatialdistributionofthemuonbeam Table4.7:Deviationstotime-independent C e correctionsduringRun-1,basedontracking simulationsandopticallatticecalculationswiththeCOSY-basedmodelofthe g -2storage ring. C e [ppb] Run-1a(60h) 0 : 24 0 : 013 Run-1b(HK) 2 : 76 0 : 011 Run-1c(9d) 3 : 68 0 : 014 Run-1d(EG) 2 : 50 0 : 012 209 duringthedatatakingperiod,themeasuredanomalousmagneticmomentofthemuonis proportionaltotheratiobetweenthe g -2frequency ! a 0 andthetotal ~ B experiencedby thebeam.Inspiteofthehighuniformityofthemagneticinthering'sstoragevolume (i.e., j ~ B j = B ˇ B y to O (10ppb)accuracy[ 39 ]),itsppm-levellocalizedvariationsalong theringazimuthcancoupletotheazimuthalbehaviorofthestoredbeam.Attheprevious muon g -2experimentatBNL,theerrorbudgetoftheir a measurementallowedtosafely calculatetheaveraged ~ B withouttheconsiderationofthebeamazimuthalvariations[ 8 ]. However,aconsiderationofsuchbeamvariationsisrequiredforthemuonweightingofthe magnetictobedeterminedtoaprecisionof10ppborless[ 46 ]andinthiswayachieve theprecisiongoalsofthemuon g -2experimentatFermilab. Forthispurpose,amethodtoperformthemuonweightingnecessarytoobtain ~ B while accountingforbeamazimuthalvariationsbasedonstrawtrackingdetectorsdata[ 38 ]and opticallatticefunctions(seeSec. 3.6.2 )hasbeendeveloped[ 106 ]andispresentedinthis section.Themethod,whichallowscomputingmuon-weightedmagneticinanorderly manner,hasbeenimplementedaspartoftheRun-1magneticanalysis.Systematic errorsassociatedwiththemethod(e.g.,detectionacceptanceandalignment)havebeen analyzedindetailanddocumentedin[ 106 ]andarenotstudiedinthisdissertation. Tobenchmarktheaforementionedmethod,amuon-weighted ~ B orig isprepared(see Sec. 4.5.1 )vianonlinearsimulationswiththemagneticintheCOSY-basedmodelfrom trolleydata,wherethebeamdistributionistrackedatseveralazimuthallocationsofthe ring.Withtheseresultsasreference,themethodtoreconstruct ~ B orig outofinformation ofthebeamtransverseintensityatoneazimuthallocation(named ~ B recon )presentedin Subsec 4.5.2 isvalidatedasshowninSec. 4.5.3 .Lastly,sensitivitystudiestounderstandthe dependenceofthemuon-weightedcalculationonspazimuthalvariationsofthe 210 storedbeamarediscussed. 4.5.1Themuon-weighted ~ B fromthesimulatedbeam Ideally,tocapturethecouplingbetweenandbeamdistortionsalongthestoragering azimuthinthemuonweightingofthemagnetic B ( x;y; ),itisweightedbythetime- andazimuth-dependentmuontransverseintensity M T ( x;y;;t )throughoutthedatataking period[ 39 ]: ~ B = Z t f t 0 Z x Z y M T ( x;y;;t ) B ( x;y; ) dxdydt 0 ; (4.33) where istheazimuthalangle, x and y theradialandverticalcoordinates, t isthetimewhere t =0coincideswithbeaminjectionintothering,andthebeamintensityisnormalized: Z t f t 0 Z x Z y M T ( x;y;;t ) dxdydt 0 =1 : (4.34) However,theazimuthalgradientsoftheopticallatticefunctions(includingclosedorbits)in theringwithratherweakverticalfocusingandtheresolutionoftrackerexperimentaldata aresmallenoughtoconvenientlyconsideradiscretesummationinstead: ~ B = 1 N N X i =1 ~ B i = 1 N N X i =1 N x X j N y X k M T x j ;y k ; i B i x j ;y k ; (4.35) where N x X j N y X k M T x j ;y k ; i =1(4.36) and N :Numberofazimuthalslices. 211 N x :Numberof x bins. N y :Numberof y bins. M T x j ;y k ; i :Binnedtransversebeamat i integratedover30 s