EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN) CERN-PH-EP/2012-271 2012/10/03 CMS-BPH-11-010 Measurement of the relative prompt production rate of χ √ c2 and χ in pp collisions at s = 7TeV c1 2 ∗ The CMS Collaboration 1 0 2 t c O 2 ] x Abstract e - p e A measurement is presented of the relative prompt production rate of χc2 and χc1 h with 4.6fb−1 of data collected by the CMS experiment at the LHC in pp collisions at √ [ s = 7TeV. The two states are measured via their radiative decays χ → J/ψ+γ, c 1 with the photon converting into an e+e− pair for J/ψ rapidity |y(J/ψ)| < 1.0 and v photon transverse momentum p (γ) > 0.5GeV/c. The measurement is given for six 5 T 7 intervalsof p (J/ψ) between7 and25GeV/c. Theresults arecompared totheoretical T 8 predictions. 0 . 0 1 2 SubmittedtotheEuropeanPhysicalJournalC 1 : v i X r a ∗SeeAppendixAforthelistofcollaborationmembers 1 1 Introduction Understandingcharmoniumproductioninhadroniccollisionsisachallengeforquantumchro- modynamics (QCD). The J/ψ production cross section measurements at the Tevatron [1, 2] were found to disagree by about a factor of 50 with theoretical color-singlet calculations [3]. Soon after, the CDF experiment reported a χ /χ cross section ratio that extended up to c2 c1 p (J/ψ) (cid:39) 10GeV/c, where p is the transverse momentum, and favored χ production over T T c1 χ [4]. The cross section ratio was also studied recently at the Large Hadron Colllider (LHC) c2 in Ref. [5]. These measurements independently suggest that charmonium production cannot beexplainedthroughrelativelysimplemodels. This paper presents a measurement of the χ /χ cross section ratio by the Compact Muon c2 c1 Solenoid (CMS) experiment at the LHC in pp collisions at a center-of-mass energy of 7TeV. Themeasurementisbasedonthereconstructionoftheχ radiativedecaystoJ/ψ+γ,withthe c lowtransversemomentumphotons(lessthan5GeV/c)beingdetectedthroughtheirconversion into electron-positron pairs. The analysis uses data collected in 2011, corresponding to a total integrated luminosity of 4.6fb−1. When estimating acceptance and efficiencies, we assume thattheχ andχ areproducedunpolarized,andwesupplythecorrectionfactorsneededto c2 c1 modifytheresultsforseveraldifferentpolarizationscenarios. Due to the extended reach in transverse momentum made possible by the LHC energies, the crosssectionratiomeasurementisexpectedtodiscriminatebetweendifferentpredictions,such as those provided by the k -factorization [6] and next-to-leading order nonrelativistic QCD T (NRQCD)[7]theoreticalapproaches. The strength of the ratio measurement is that most theoretical uncertainties cancel, including thequarkmasses, thevalueofthestrongcouplingconstant α , aswellasexperimentaluncer- s taintiesonquantitiessuchasintegratedluminosity,triggerefficiencies,and,inpart,reconstruc- tionefficiency. Therefore,thisratiocanberegardedasanimportantreferencemeasurementto test the validity of various theoretical quarkonium production models. With this paper, we hopetoprovidefurtherguidanceforfuturecalculations. 2 CMS detector A detailed description of the CMS apparatus is given in Ref. [8]. Here we provide a short summaryofthedetectorsrelevantforthismeasurement. The central feature of the CMS apparatus is a superconducting solenoid of 6m internal diam- eter. Within the field volume are the silicon pixel and strip tracker, the crystal electromag- netic calorimeter and the brass/scintillator hadron calorimeter. Muons are measured in gas- ionization detectors embedded in the steel return yoke. In addition to the barrel and endcap detectors,CMShasextensiveforwardcalorimetry. Theinnertrackermeasureschargedparticleswithinthepseudorapidityrange|η| < 2.5,where η = −ln[tan(θ/2)], and θ is the polar angle measured from the beam axis. It consists of 1440 silicon pixel and 15148 silicon strip detector modules. In the central region, modules are ar- rangedin13measurementlayers. Itprovidesanimpactparameterresolutionof∼15µm. Muonsaremeasuredinthepseudorapidityrange|η| < 2.4,withdetectionplanesmadeusing three technologies: drift tubes, cathode strip chambers, and resistive plate chambers. Match- ing the muons to the tracks measured in the silicon tracker results in a transverse momentum resolutionbetween1and1.5%,for p valuesupto50GeV/c. T 2 4 Eventreconstructionandselection Thefirstlevel(L1)oftheCMStriggersystem,composedofcustomhardwareprocessors,uses informationfromthecalorimetersandmuondetectorstoselectthemostinterestingevents. The high-level trigger (HLT) processor farm further decreases the event rate from around 100kHz to around 300Hz, before data storage. The rate of HLT triggers relevant for this analysis was intherange5–10Hz. Weanalyzedabout60millionsuchtriggers. 3 Experimental method We select χ and χ candidates by searching for their radiative decays into the J/ψ+γ final c1 c2 state, with the J/ψ decaying into two muons. The χ has too small a branching fraction into c0 thisfinalstatetoperformausefulmeasurement,butweconsideritinthemodelingofthesig- nal lineshape. Given the small difference between the J/ψ mass, 3096.916±0.011MeV/c2, and the χ and χ masses, 3510.66±0.07MeV/c2 and 3556.20±0.09MeV/c2, respectively [9], the c1 c2 detector must be able to reconstruct photons of low transverse momentum. In addition, ex- cellentphotonmomentumresolutionisneededtoresolvethetwostates. Inthecenter-of-mass frameoftheχ states,thephotonhasanenergyof390MeVwhenemittedbyaχ and430MeV c c1 whenemittedbyaχ . Thisresultsinmostofthephotonshavinga p inthelaboratoryframe c2 T smaller than 6GeV/c. The precision of the cross section ratio measurement depends crucially ontheexperimentalphotonenergyresolution,whichmustbegoodenoughtoseparatethetwo states. Averyaccuratemeasurementofthephotonenergyisobtainedbymeasuringelectron- positronpairsoriginatingfromaphotonconversioninthebeampipeortheinnerlayersofthe silicon tracker. The superior resolution of this approach, compared to a calorimetric energy measurement, comes at the cost of a reduced yield due to the small probability for a conver- siontooccurintheinnermostpartofthetrackerdetectorand,moreimportantly, bythesmall reconstruction efficiency for low transverse momentum tracks whose origin is displaced with respect to the beam axis. Nevertheless, because of the high χ production cross section at the c LHC,theuseofconversionsleadstothemostpreciseresult. For each χ candidate, we evaluate the mass difference ∆m = m − m between the c1,2 µµγ µµ dimuon-plus-photon invariant mass, m , and the dimuon invariant mass, m . We use the µµγ µµ quantity Q = ∆m+m , where m is the world-average mass of the J/ψ from Ref. [9], as J/ψ J/ψ a convenient variable for plotting the invariant-mass distribution. We perform an unbinned maximum-likelihoodfittotheQspectrumtoextracttheyieldofpromptχ andχ asafunc- c1 c2 tion of the transverse momentum of the J/ψ. A correction is applied for the differing accep- tances for the two states. Our results are given in terms of the prompt production ratio R , p definedas σ(pp → χ +X)B(χ → J/ψ+γ) N ε R ≡ c2 c2 = χc2 · 1, p σ(pp → χ +X)B(χ → J/ψ+γ) N ε c1 c1 χc1 2 where σ(pp → χ + X) are the χ production cross sections, B(χ → J/ψ+γ) are the χ c c c c branching fractions, N are the number of candidates of each type obtained from the fit, and χi ε /ε is the ratio of the efficiencies for the two χ states. The branching fractions B(χ → 1 2 c c1,2 J/ψ+γ),takenfromRef.[9],arealsousedtocalculatetheratioofproductioncrosssections. 4 Event reconstruction and selection In order to select χ signal events, a dimuon trigger is used to record events containing the c decay J/ψ → µµ. The L1 selection requires two muons without an explicit constraint on 3 their transverse momentum. At the HLT, opposite-charge dimuons are reconstructed and the dimuonrapidityy(µµ)isrequiredtosatisfy|y(µµ)| < 1.0,whilethedimuon p mustexceeda T thresholdthatincreasedfrom6.5to10GeV/casthetriggerconfigurationevolvedtocopewith the instantaneous luminosity increase. Events containing dimuon candidates with invariant massfrom2.95to3.25GeV/c2 arerecorded. Ourdatasampleconsistsofeventswheremultiple pp interactions occur. At each bunch crossing, an average of six primary vertices is recon- structed,oneofthemrelatedtotheinteractionthatproducestheχ inthefinalstate,theothers c relatedtosoftercollisions(pileup). In the J/ψ selection, the muon tracks are required to pass the following criteria. They must have at least 11 hits in the tracker, with at least two in the pixel layers, to remove background from decays-in-flight. The χ2 per degree of freedom of the track fit must be less than 1.8. To remove background from cosmic-ray muons, the tracks must intersect a cylindrical volume of radius 4cm and total length 70cm, centered at the nominal interaction point and with its axis parallel to the beam line. Muon candidate tracks are required to have p > 3.3GeV/c, T |η| ≤ 1.3 and match a well-reconstructed segment in at least one muon detector [10]. Muons with opposite charges are paired. The two muon trajectories are fitted with a common vertex constraint, and events are retained if the fit χ2 probability is larger than 1%. If more than one muonpairisfoundinanevent,onlythepairwiththelargestvertexχ2 probabilityisselected. For the final χ and χ selection, a dimuon candidate must have an invariant mass between c1 c2 3.0and3.2GeV/c2 and|y| < 1.0. In order to restrict the measurement to the prompt J/ψ signal component, the pseudo-proper decaylengthoftheJ/ψ ((cid:96) ),definedas (cid:96) = L ·m /p (J/ψ),where L isthemostprob- J/ψ J/ψ xy J/ψ T xy able transverse decay length in the laboratory frame [11], is required to be less than 30 µm. In the region (cid:96) < 30µm, we estimate, from the observed (cid:96) distribution, a contamination J/ψ J/ψ ofthenonpromptcomponent(originatingfromthedecaysofBhadrons)ofabout0.7%,which hasanegligibleimpactonthetotalsystematicuncertainty. To reconstruct the photon from radiative decays, we use the tracker-based conversion recon- structiondescribedinRefs.[12–14]. Wesummarizethemethodhere,mentioningthefurtherre- quirementsneededtospecializetheconversionreconstructionalgorithmtotheχ case. Theal- c gorithmreliesonthecapabilityofiterativetrackingtoefficientlyreconstructdisplacedandlow transverse momentum tracks. Photon conversions are characterized by an electron-positron pairoriginatingfromacommonvertex. Thee+e− invariantmassmustbeconsistentwithzero within its uncertainties and the two tracks are required to be parallel at the conversion point andtoseparatefromeachotheronlyinthetransverseplane. Opposite-sign track pairs are first required to have more than four hits and a normalized χ2 less than 10. Then the reconstruction algorithm exploits the conversion-pair signature to dis- tinguishbetweengenuineandmisidentifiedbackgroundpairs. Informationfromthecalorime- ters is not used for conversion reconstruction in our analysis. The primary pp collision vertex associated with the photon conversion, see below, is required to lie outside both track helices. Helicesprojectedontothetransverseplaneformcircles;wedefined asthedistancebetween m thecentersofthetwocirclesminusthesumoftheirradii. Thevalueofd isnegativewhenthe m twoprojectedtrajectoriesareintersecting. Werequirethecondition−0.25 < d < 1.0cmtobe m satisfied. In order to reduce the contribution of misidentified conversions from low-momentum dis- placedtracksthatareartificiallypropagatedbacktothesilicontracker,thetwocandidatecon- versiontracksmusthaveoneoftheirtwoinnermosthitsinthesamesilicontrackerlayer. 4 5 Acceptanceandefficiencies Thedistancealongthebeamlinebetweentheextrapolationofeachconversiontrackcandidate and the nearest reconstructed event vertex must be less than five times its estimated uncer- tainty. Moreover, among the two event vertices closest to each track along the beam line, at leastonevertexmustbeincommon A reconstructed primary vertex is assigned to the reconstructed conversion by projecting the photon momentum onto the beamline and choosing the closest vertex along the beam direc- tion. If the value of the distance is larger than five times its estimated uncertainty, the photon candidateisrejected. Theprimaryvertexassociatedwiththeconversionisrequiredtobecompatiblewiththerecon- structedJ/ψvertex. Thisrequirementisfulfilledwhenthethree-dimensionaldistancebetween thetwoverticesiscompatiblewithzerowithinfivestandarddeviations. Furthermore,acheck ismadethatneitherofthetwomuontracksusedtodefinetheJ/ψ vertexisusedasoneofthe conversiontrackpair. Thee+e−trackpairssurvivingtheselectionarethenfittedtoacommonvertexwithakinematic vertex fitter that constrains the tracks to be parallel at the vertex in both the transverse and longitudinalplanes. Thepairisretainedifthefitχ2 probabilityisgreaterthan0.05%. Ifatrack is shared among two or more reconstructed conversions, only the conversion with the larger vertexχ2 probabilityisretained. Only reconstructed conversions with transverse distance of the vertex from the center of the mean pp collision position larger than 1.5cm are considered. This requirement suppresses backgroundscausedbytrackpairsoriginatingfromtheprimaryeventvertexthatmightmimic a conversion, such as from π0 Dalitz decay, while retaining photon conversions occurring withinthebeampipe. Finally, each conversion candidate is associated with every other conversion candidate in the event, and with any photon reconstructed using calorimeter information. Any pairs of con- versionsorconversionplusphotonwithaninvariantmassbetween0.11and0.15GeV/c2,cor- responding to a two-standard-deviation window around the π0 mass, is rejected. We have verifiedthattheπ0 rejectionrequirement,whileeffectivelyreducingthebackground,doesnot affectthe R measurementwithinitsuncertainties. p Converted photon candidates are required to have p > 0.5GeV/c, while no requirement is T imposedonthepseudorapidityofthephoton. Thedistributionofthephotonconversionradiusforχ candidatesisshowninFig.1. Thefirst c peakcorrespondstothebeampipeandfirstpixelbarrellayer,thesecondandthirdpeakscorre- spondtothetwooutermostpixellayers,whiletheremainingfeaturesatradiilargerthan20cm are due to the four innermost silicon strip layers. The observed distribution of the conversion radiusagreeswellwithMCsimulations[14],whichindicatesagoodknowledgeofthematerial inthetrackingvolume. 5 Acceptance and efficiencies IntheevaluationofR ,wemusttakeintoaccountthepossibilitythatthegeometricacceptance p andthephotonreconstructionefficienciesarenotthesameforχ andχ . c1 c2 In order to determine the acceptance correction, a Monte Carlo (MC) simulation sample of equal numbers of χc1 and χc2 has been used. This sample was produced using a PYTHIA [15] single-particlesimulationinwhicha χ or χ isgeneratedwithatransversemomentumdis- c1 c2 5 m m 7000 5 CMS per 6000 pp, s = 7 TeV s on L = 4.6 fb-1 si 5000 er v n Co 4000 3000 2000 1000 0 0 5 10 15 20 25 30 35 40 45 50 Conversion radius [cm] Figure1: Distributionoftheconversionradiusfortheχ photoncandidates. c tribution produced from a parameterized fit to the CMS measured ψ(2S) spectrum [16]. The useoftheψ(2S)spectrumismotivatedbytheproximityoftheψ(2S)masstothestatesunder examination. TheimpactofthischoiceisdiscussedinSection7. Both χ statesinthesimulationareforcedtodecaytoJ/ψ+γ isotropicallyintheirrestframe, c i.e., assumingtheyareproducedunpolarized. Wediscusslatertheimpactofthisassumption. The decay products are then processed through the full CMS detector simulation, based on GEANT4 [17, 18], and subjected to the trigger emulation and the full event reconstruction. In ordertoproducethemostrealisticsampleofsimulated χ decays,digitizedsignalsfromMC- c simulated inelastic pp events are mixed with those from simulated signal tracks. The number of inelastic events to mix with each signal event is sampled from a Poisson distribution to accuratelyreproducetheamountofpileupinthedata. Theefficiencyratioε /ε fordifferentJ/ψtransversemomentumbinsisdeterminedusing: 1 2 ε Nrec Nrec 1 = χc1 / χc2 , ε2 Nχgce1n Nχgce2n where Ngen is the number of χ candidates generated in the MC simulation within the kine- c matic range |y(J/ψ)| < 1.0, p (γ) > 0.5GeV/c, and Nrec is the number of candidates recon- T structedwiththeselectionabove. TheresultingvaluesareshowninTable1,wheretheuncer- tainties are statistical only and determined from the MC sample assuming binomial distribu- tions. The increasing trend of ε /ε is expected, because p (J/ψ) is correlated with the p of 1 2 T T thephoton,andathigherphoton p ourconversionreconstructionefficiencyisapproximately T constant. Therefore, efficiencies for the χ and the χ are approximately the same at high c1 c2 p (J/ψ). T This technique also provides an estimate of the absolute χ reconstruction efficiency, which c is given by the product of the photon conversion probability, the χ selection efficiency, and, c mostimportantly,theconversionreconstructionefficiency,whichcorrespondstothedominant contribution. Thisproductvariesasafunctionof p (γ),andgoesfrom4×10−4 at0.5GeV/cto T around10−2 at4GeV/c,whereitsaturates. 6 7 Systematicuncertainties Table 1: Ratio of efficiencies ε /ε as a function of the J/ψ transverse momentum from MC 1 2 simulation. Theuncertaintiesarestatisticalonly. p (J/ψ)[GeV/c] ε /ε T 1 2 7–9 0.903±0.023 9–11 0.935±0.019 11–13 0.945±0.021 13–16 0.917±0.022 16–20 0.981±0.031 20–25 1.028±0.049 6 Signal extraction We extract the numbers of χ and χ events, N and N , respectively, from the data by c1 c2 χc1 χc2 performing an unbinned maximum-likelihood fit to the Q spectrum in various ranges of J/ψ transversemomentum. Because of the small intrinsic width of the χ states we are investigating, the observed signal c shape is dominated by the experimental resolution. The signal probability density function (PDF)isderivedfromtheMCsimulationdescribedinSection5,andismodeledbythesuper- position of two double-sided Crystal Ball functions [19] for the χ and χ and a single-sided c1 c2 CrystalBallfunctionfortheχ . Eachdouble-sidedCrystalBallfunctionconsistsofaGaussian c0 core with exponential tails on both the high- and low-mass sides. We find this shape to pro- videanaccurateparameterizationofthe QspectraderivedfromMCsimulation. Whenfitting the data, we fix all the parameters of the Crystal Ball function to the values that best fit our MC simulation and use a maximum-likelihood approach to derive N and N , which are χc1 χc2 theintegralsofthePDFsforthetworesonances. Thebackgroundismodeledbyaprobability distributionfunctiondefinedas N (Q) = (Q−q )α1 ·e(Q−q0)·β1, bkg 0 whereα and β arefreeparametersinthefit,andq issetto3.2GeV/c2. 1 1 0 In Fig. 2 we show the Q distribution for two different ranges, 11 < p (J/ψ) < 13GeV/c (left) T and 16 < p (J/ψ) < 20GeV/c (right). This procedure is repeated for several ranges in the T transverse momentum of the J/ψ in order to extract N and N in the corresponding bin. χc1 χc2 TheresultsareshowninTable2,wherethereporteduncertaintiesarestatisticalonly. Table 2: Numbers of χ and χ events extracted from the maximum-likelihood fit, and the c1 c2 ratioofthetwovalues. Uncertaintiesarestatisticalonly. p (J/ψ)[GeV/c] N N N /N T χc1 χc2 χc2 χc1 7–9 618±31 315±24 0.510±0.049 9–11 1680±49 788±37 0.469±0.027 11–13 1819±51 819±38 0.451±0.025 13–16 1767±51 851±39 0.482±0.027 16–20 1269±43 487±30 0.384±0.028 20–25 642±31 236±22 0.368±0.040 7 Systematic uncertainties Several types of systematic uncertainties are addressed. In particular, we investigate possible effects that could influence the measurement of the numbers of χ and χ from data, the c1 c2 7.1 Uncertaintyfromthemassfitandχ andχ counting 7 c1 c2 2 400 2 c c V/ CMS V/250 CMS Me350 pp, s = 7 TeV Me pp, s = 7 TeV s/5 300 L = 4.6 fb-1 s/5 200 L = 4.6 fb-1 nt nt Eve250 1113 GGeeVV//cc <> ppT((JJ//yy )) Eve150 1260 GGeeVV//cc <> ppT((JJ//yy )) 200 T T 150 100 100 50 50 0 0 3.2 3.4 3.6 3.8 3.2 3.4 3.6 3.8 m - m + m [GeV/c2] m - m + m [GeV/c2] m m g m m J/y m m g m m J/y Figure2: ThedistributionofthevariableQ = m −m +m forχ candidateswithp (J/ψ) µµγ µµ J/ψ c T rangesshowninthefigures. Thelineshowsthefittothedata. evaluation of ε /ε from the MC simulation, and the derivation of the R ratio. In Table 3 the 1 2 p various sources of systematic uncertainties and their contributions to the total uncertainty are summarized. Thefollowingsubsectionsdescribehowthevariouscontributionsareevaluated. Table3: Relativesystematicuncertaintieson R fordifferentrangesofJ/ψ transversemomen- p tumfromdifferentsourcesandthetotaluncertainty. p (J/ψ)range[GeV/c] 7–9 9–11 11–13 13–16 16–20 20–25 T Sourceofuncertainty Systematicuncertainty(%) Backgroundshape 1.4 1.5 0.9 1.2 1.8 2.4 Signalshape 1.4 3.0 1.1 1.5 1.5 2.3 Simulationsamplesize 2.6 2.0 2.2 2.4 3.1 4.8 Choiceof p (χ )spectrum 4.5 3.7 2.9 1.9 0.6 1.1 T c Totaluncertainty 5.5 5.4 3.9 3.6 4.0 5.9 7.1 Uncertainty from the mass fit and χ and χ counting c1 c2 Themeasurementoftheratio N /N couldbeaffectedbythechoiceofthefunctionalform χc2 χc1 used for the maximum-likelihood fit. The use of an alternative background parameterization, afourth-orderpolynomial,resultsinsystematicallyhighervaluesoftheratio N /N ,while χc2 χc1 keeping the overall fit quality as high as in the default procedure. From the difference in the numbersofsignaleventsusingthetwobackgroundparameterizations,weassignthesystem- aticuncertaintyfromthebackgroundmodelingshowninTable3. We evaluate the systematic uncertainty related to the parameterization of the signal shape by varyingtheparametersderivedfromtheMCsimulationwithintheiruncertainties. Theresults fluctuate within 1–3% in the various transverse momentum ranges. We assign the systematic uncertaintiesfromthissource,asshowninTable3. The method to disentangle and count the χc1 and χc2 states is validated by using a PYTHIA MC simulation sample of inclusive J/ψ events, including those from χ decay, produced in c pp collisions and propagated through the full simulation of the detector. The ratio N /N χc2 χc1 derived from the fit to the Q distribution of the reconstructed candidates in the simulation 8 7 Systematicuncertainties is consistent with the actual number of χ events contributing to the distribution, within the c statistical uncertainty, for all J/ψ momentum ranges. Therefore, we do not assign any further systematicuncertaintyonthedeterminationof N /N . χc2 χc1 The stability of our analysis as a function of the number of primary vertices in the event has beeninvestigated. Thenumberofχ candidatesperunitofintegratedluminosity,oncetrigger c conditionsaretakenintoaccount,isfoundtobeindependentoftheinstantaneousluminosity, within the statistical uncertainties. In addition, the measured ratio N /N is found to be χc2 χc1 constant as a function of the number of primary vertices in the event, within the statistical uncertainties. Thus,nosystematicuncertaintyduetopileupisincludedinthefinalresults. 7.2 Uncertainty on the ratio of efficiencies Thestatisticaluncertaintyonthemeasurementofε /ε fromthesimulation,owingtothefinite 1 2 sizeoftheMCsample,istakenasasystematicuncertainty,asshowninTable3. Since the analysis relies on photon conversions, the effect of a possible incorrect simulation of thetrackerdetectormaterialisestimated. Twomodifiedmaterialscenarios,i.e.,specialdetector geometriespreparedforthispurpose,inwhichthetotalmassofthesilicontrackervariesbyup to5%fromthereferencegeometry,areusedtoproducenewMCsimulationsamples[20]. With these models, local variations of the radiation length with respect to the reference simulation canbeaslargeas+8%and−3%. Nosignificantdifferenceintheratioofefficienciesisobserved andthecorrespondingsystematicuncertaintyistakentobenegligible. Severalchoicesofthegenerated p (χ )spectrumareinvestigated. Inparticular,theuseofthe T c measuredJ/ψ spectrum[11]givesvaluesthatarecompatiblewiththedefault ψ(2S) spectrum usedforthefinalresult. Thechoiceofthespectrumaffectsthevaluesofε /ε onlyinasmuchas 1 2 weperformanaveragemeasurementineachbinof p (J/ψ),andthesizeofthesebinsisfinite. T We choose to assign a conservative systematic uncertainty by comparing the values of ε /ε 1 2 obtained with the ψ(2S) spectrum with those obtained in the case where the p (χ ) spectrum T c istakentobeconstantineach p bin. Thecorrespondingsystematicuncertaintiesaregivenin T Table3. 7.3 χ polarization c Thepolarizationsoftheχ andχ areunknown. Efficienciesareestimatedundertheassump- c1 c2 tionthatthetwostatesareunpolarized. Iftheχ statesarepolarized,theresultingphotonan- c gular distribution and transverse momentum distributions will be affected. This can produce achangeinthephotonefficiencyratioε /ε . 1 2 In order to investigate the impact of different polarization scenarios on the ratio of the effi- ciencies,wereweighttheunpolarizedMCdistributionstoreproducethetheoreticalχ angular c distributions [21,22]fordifferentχ polarizations. Wemeasuretheefficiencyε /ε fortheχ c 1 2 c1 being unpolarized or with helicity m = 0,±1, in combination with the χ being unpolar- χc1 c2 ized or having helicity m = 0,±2 in both the helicity and Collins–Soper [23] frames. The χc2 ratio of efficiencies for the cases involving m = ±1 is between the cases with m = 0 and χc2 χc2 m = ±2. Tables4and5givetheresulting ε /ε valuesforeachpolarizationscenarioindif- χc2 1 2 ferentJ/ψtransversemomentumbinsforthetwoframes,relativetothevalueoftheratioforthe unpolarized case. These tables, therefore, provide the correction that should be applied to the defaultvalueofε /ε ineachpolarizationscenarioandeachrangeoftransversemomentum. 1 2