Running of the top quark mass from proton-proton collisions at √s=13 TeV

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Physics Letters B

www.elsevier.com/locate/physletb

Running of the top quark mass from proton-proton collisions at

√ s = ^{13 TeV}

.The CMS Collaboration

^

CERN,Switzerland

a r t i c l e i n f o a b s t ra c t

Articlehistory:

Received19September2019

Receivedinrevisedform26January2020 Accepted27January2020

Availableonline10February2020 Editor:M.Doser

Keywords:

CMS Physics Topquarkmass QCD

Renormalization

Therunningofthetopquarkmassisexperimentallyinvestigatedforthefirsttime.Themassofthetop quarkinthemodifiedminimalsubtraction(MS)renormalizationschemeisextractedfromacomparison ofthe differential topquark-antiquark (t¯t)cross sectionas afunctionoftheinvariant mass ofthe t¯t systemto next-to-leading-order theoreticalpredictions. The differentialcross sectionisdeterminedat the partonlevelby meansofamaximum-likelihoodfitto distributionsoffinal-state observables.The analysisisperformedusingt¯t candidateeventsinthee^±μ^∓channelinproton-protoncollisiondataata centre-of-massenergyof13 TeV recordedbytheCMSdetectorattheCERNLHCin2016,corresponding toanintegratedluminosityof35.9 fb⁻¹.Theextractedrunningisfoundtobecompatiblewiththescale dependencepredictedbythecorrespondingrenormalizationgroupequation.Inthisanalysis,therunning isprobeduptoascaleoftheorderof1 TeV.

©^{2020TheAuthor(s).PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense} (http://creativecommons.org/licenses/by/4.0/).FundedbySCOAP³.

1. Introduction

Beyondleading order inperturbation theory,the fundamental parametersofthequantumchromodynamics(QCD)Lagrangian,i.e.

thestrongcouplingconstant

α

S andthequarkmasses,aresubject to renormalization. As a result, these parameters depend on the scaleatwhichthey areevaluated.Theevolutionof

α

S andofthe quarkmassesasafunctionofthe scale,commonlyreferredtoas

“running”,isdescribedbyrenormalizationgroupequations(RGEs).

The running of

α

S was experimentally verified on a wide range ofscalesusing jetproductionin electron-proton,positron-proton, electron-positron,proton-antiproton,andproton-proton(pp)colli- sions,assummarized,e.g. inRefs. [1,2].Todeterminetherunning, the value of

α

S evaluated at an arbitrary reference scale is ex- tractedinbinsofaphysicalenergyscale Q andthenconvertedto

α

S(Q)usingthe corresponding RGE [2]. Thevalidity ofthis pro- cedure lies inthe fact that, ina calculation, the renormalization scaleis normallyidentified withthephysical energyscale ofthe process. The sameprocedure can be used to determine the run- ningofthemassofaquark. Inthemodified minimalsubtraction (MS)renormalizationscheme,thedependenceofaquark massm onthescale

μ

isdescribedbytheRGE

 E-mailaddress:cms-publication-committee-chair@cern.ch.

μ

^2dm

⁽ μ )

d

μ

²

^{= −} γ ( α

S

( μ ))

m

( μ ),

(1) where

γ

(

α

S(

μ

)) is the mass anomalous dimension, which is knownup to five-looporder inperturbative QCD [3,4].The solu- tionofEq. (1) canbeused toobtainthequark massatanyscale

μ

from the mass evaluated at an initial scale

μ

0. The running ofthe b quark masswas demonstrated [5] usingdata fromvari- ous experimentsatthe CERNLEP [6–9],SLACSLC [10],andDESY HERA [11] colliders. Measurementsof charm quark pair produc- tion in deepinelastic scatteringat theDESY HERA were used to determinetherunningofthecharmquarkmass [12].Thesemea- surementsrepresentapowerfultestofthevalidityofperturbative QCD. Furthermore, RGEs can be modified by contributions from physics beyond thestandard model, e.g. inthe context of super- symmetrictheories [13].

ThisLetterdescribesthefirstexperimentalinvestigationofthe runningofthetop quarkmass,mt,asdefinedintheMS scheme.

Therunningofm_t isextractedfroma measurementofthediffer- ential top quark-antiquarkpairproductioncross section,

σ

_t¯t,asa functionoftheinvariant massofthe t¯t system,m_t¯t.Thedifferen- tialcross section,d

σ

_t¯t/dm_t¯t,is determinedatthe partonlevelby means of a maximum-likelihoodfit to distributions of final-state observablesusingt¯t candidateeventsinthee^±μ^{∓ final state,ex-} tendingthemethoddescribedinRef. [14] tothecaseofadifferen- tialmeasurement.Thismethodallowsthedifferentialcrosssection to be constrained simultaneously with the systematic uncertain- https://doi.org/10.1016/j.physletb.2020.135263

0370-2693/©^{2020TheAuthor(s).PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense}(http://creativecommons.org/licenses/by/4.0/).Fundedby SCOAP³.

ties. In this analysis, the parton level is definedbefore radiation fromthepartonshower,whichallowsforadirectcomparisonwith fixed-ordertheoreticalpredictions.Themeasurementisperformed using pp collision dataat √

s=¹³TeV recorded by the CMSde- tector at the CERN LHC in 2016, corresponding to an integrated luminosity of35.9 fb⁻¹.The runningmass,m_t(

μ

),is extractedat next-to-leadingorder(NLO) inQCD asafunction ofm_t¯t by com- paringfixed-order theoreticalpredictionsatNLOtothemeasured d

σ

_t¯t/dm_t¯t.Therunningofmt isprobeduptoa scaleoftheorder of1 TeV.

2. TheCMSdetectorandMonteCarlosimulation

The central feature of the CMS apparatus is a superconduct- ing solenoidof 6 m internal diameter, providinga magnetic field of3.8 T. Withinthe solenoidvolume area siliconpixel andstrip tracker,aleadtungstatecrystalelectromagneticcalorimeter(ECAL), andabrass andscintillatorhadron calorimeter,eachcomposedof abarrelandtwoendcapsections.Forwardcalorimetersextendthe pseudorapidity (

η

) coverage provided by the barrel and endcap detectors.Muonsaredetected ingas-ionizationchambersembed- dedinthesteelflux-returnyokeoutsidethesolenoid. Atwo-level triggersystemselects eventsofinterestforanalysis [15].A more detaileddescriptionoftheCMSdetector,togetherwithadefinition ofthecoordinatesystemusedandtherelevantkinematicvariables, canbefoundinRef. [16].

The particle-flow (PF) algorithm [17] aims to reconstruct and identify electrons, muons, photons, charged and neutral hadrons inan event, withan optimizedcombination ofinformation from thevariouselementsoftheCMSdetector.Theenergyofelectrons isdetermined from acombination ofthe electron momentumat the primary interaction vertex asdetermined by the tracker,the energyofthecorresponding ECALcluster, andthe energysumof all bremsstrahlung photons spatially compatible with originating from the electron track [18]. The momentum of muons is ob- tained from the curvature of the corresponding track [19]. Jets are reconstructed from the PFcandidates using the anti-kT clus- teringalgorithmwithadistanceparameterof0.4 [20,21],andthe jet momentum isdetermined as thevectorial sumofall particle momentain thejet. Themissingtransverse momentum vectoris computedasthenegative vector sumof thetransversemomenta (pT) ofallthePFcandidatesinanevent.Jetsoriginatingfromthe hadronization ofb quarks(b jets) are identified (b tagged) using the combined secondary vertex [22] algorithm, using a working pointthatcorrespondstoanaverageb taggingefficiencyof41%for simulatedt¯t events,andanaveragemisidentificationprobabilityof 0.1%and2.2%forlight-flavourjetsandc jets,respectively [22].

Inthisanalysis, the sameMonte Carlo(MC) simulationsas in Ref. [14] are used.Inparticular,t¯t, tW,andDrell–Yan(DY) events are simulated using the powheg v2 [23–28] NLO MC generator interfaced to pythia 8.202 [29] for the modelling of the parton showerandusingtheCUETP8M2T4underlyingeventtune [30,31].

In the simulation, the proton structure is described by means ofthe NNPDF3.0 [32] parton distributionfunction (PDF) set.The largest background contributions are represented by tW and DY production. Other background processes include W+jets produc- tion and diboson events, while the contribution from QCD mul- tijet production is found to be negligible. Contributions from all backgroundprocessesareestimatedfromsimulationandarenor- malizedtotheirpredictedcrosssection.FurtherdetailsontheMC simulationofthebackgroundscanbefoundinRef. [14].

3. Eventselectionandsystematicuncertainties

Eventsare collected usingacombinationoftriggerswhichre- quire eitherone electron with p_T>12GeV and one muon with

pT>23GeV,oroneelectronwithpT>23GeV andonemuonwith pT>8GeV,orone electronwith pT>27GeV,orone muonwith p_T>24GeV.In the analysis, tight isolation requirements are ap- pliedtoelectronsandmuonsbasedontheratioofthescalarsum of the pT ofneighbouring PF candidates to the pT of the lepton candidate. Events are then required to contain atleast one elec- tron andone muon ofopposite electriccharge with pT>25GeV for the leading and pT>20GeV for the subleading lepton, and

|

η

|<2.4.Thiskinematicselection definesthevisiblephasespace.

Ineventswithmorethantwoleptons,thetwoleptonsofopposite charge withthe highest pT are used. Jets with pT>30GeV and

|

η

|<2.4 areconsidered,butnorequirementonthenumberofre- constructedjetsorb-taggedjetsisimposed.Furtherdetailsonthe eventselectioncanbefoundinRef. [14].

In events with at leasttwo jets, the invariant mass of the t¯t systemisestimatedbymeans ofthekinematicreconstructional- gorithmdescribedinRef. [33].Thereconstructedinvariantmassis indicated with m^reco_t¯t . The kinematicreconstruction algorithm ex- aminesallpossiblecombinationsofreconstructedjetsandleptons andsolves asystemofequationsundertheassumptions that the invariant massofthereconstructedW bosonis80.4 GeV andthat themissingtransversemomentum originatessolelyfromthe two neutrinos coming from the leptonic decays of the W bosons. In addition, the kinematic reconstruction algorithm requires an as- sumptionon thevalue ofthe topquark mass,m^kin_t .Any possible bias dueto the choice of this value is avoided by incorporating thedependenceonm^kin_t inthefitdescribedinSection 4.Toesti- matethisdependence,thekinematicreconstructionandtheevent selection arerepeatedwiththreedifferentchoicesofm^kin_t , corre- sponding to169.5, 172.5,and175.5 GeV,andthe topquark mass usedintheMCsimulation,m^MC_t ,isvariedaccordingly.Theparam- eterm^kin_t =^mMCt isthentreatedasafreeparameterofthefit.

The sourcesofsystematicuncertainties areclassifiedasexper- imental and modelling uncertainties. Experimental uncertainties arerelatedtothecorrectionsappliedtotheMCsimulation.These includeuncertaintiesassociatedwithtriggerandleptonidentifica- tion efficiencies, jet energy scale [34] andresolution [35], lepton energy scales, b tagging efficiencies [22], and the uncertainty in theintegratedluminosity [36].Modellinguncertaintiesarerelated tothesimulationofthet¯t signal,andincludematrix-elementscale variationsinthe powheg simulation [37,38],scalevariationsinthe partonshower [31],variations inthematchingscalebetweenthe matrix element andthe parton shower [30], uncertainties in the underlying eventtune [30], thePDFs [39], the B hadron branch- ing fractionandfragmentationfunction [40,41], anduncertainties related to the choice of the colour reconnection model [42,43].

Furthermore,asinpreviousCMSanalyses,e.g. [14,44,45],anuncer- taintythataccountsfortheobserveddifferenceintheshapeofthe top quark pT distributionbetweendataandsimulation [33,46,47]

is applied.The dependenceon thetop quark widthhas beenin- vestigated andwas found to be negligible. Other sources of un- certainty includethe modellingof the additional pp interactions withinthesameornearbybunchcrossingsandthenormalization of background processes. Forthe latter, an uncertainty of 30% is assignedtothenormalizationofeachbackgroundprocess.Further details onthesourcesofsystematicuncertaintiesandtheconsid- eredvariationscanbefoundinRef. [14].

The simulated t¯t sample is split into four subsamples corre- sponding to bins of m_t¯t at the parton level. Each subsample is treatedasan independentsignalprocess,representingthet¯t pro- ductionatthescale

μ

k,whichischosentobethecentre-of-gravity ofbink,definedasthemeanvalueofm_t¯t inthatbin.Thesubsam- plecorresponding tothebink isdenotedwith“Signal(

μ

_k)”.The m_t¯t binboundaries,thecorrespondingfractionofsimulatedevents in each bin,andthe representative scales

μ

k are summarizedin

Table 1

The mt¯t bin boundaries, thecorresponding fraction of eventsinthe powheg simulation,andtherepresentative scaleμk.

Bin mt¯t[GeV] Fraction [%] μk[GeV]

1 <420 30 384

2 420–550 39 476

3 550–810 24 644

4 >810 7 1024

Fig. 1. Distributionofm^reco_t¯t afterthefittothedata,withthesamebinningasused inthefit.Thehatchedbandcorrespondstothetotaluncertaintyinthepredicted yields,includingthe contributionfromm^MCt (m^MCt )andallcorrelations.Thet¯t MCsampleissplitintofoursubsamples,denotedwith“Signal(μk)”,corresponding tobinsofm_t¯t atthepartonlevel.Thefirstand lastbinscontainalleventswith m^reco_t¯t <420GeV andm^reco_t¯t >810GeV,respectively.

Table1,wherethevaluesareestimatedfromthenominal powheg simulation.Thewidthofeachbin,m^k_t¯t,ischosentakingintoac- counttheresolutioninm^reco_t¯t .Fig.1showsthedistributionofm^reco_t¯t afterthefittothedata,whichisdescribedinthenextsection.

4. Fitprocedureandcrosssectionresults

Thedifferentialt¯t crosssectionatthepartonlevelismeasured by means of a maximum-likelihood fit to distributions of final- state observables where the systematic uncertainties are treated asnuisanceparameters.Inthelikelihood,thenumberofeventsin eachbinofanydistributionoffinal-stateobservablesisassumedto followaPoissondistribution.With

σ

_t⁽_¯t^μk)= (^d

σ

_t¯t/dm_t¯t)m^k_t¯t being thetotalt¯t crosssectioninthebink ofm_t¯t,theexpectednumber ofeventsinthebini ofanyoftheconsideredfinal-statedistribu- tions,denotedwith

ν

i,canbewrittenas

ν

i

=



4 k=¹

s^k_i

( σ

_t⁽_¯t^μk)

,

m^MC_t

,  λ) + 

j

b_i^j

(

m^MC_t

,  λ).

(2)

Here, s^k_i indicates theexpected numberoft¯t events inthe bink ofm_t¯t anddependson

σ

_t⁽_¯t^μk),m^MC_t ,andthenuisanceparametersλ. Similarly,b_i^jrepresentstheexpectednumberofbackgroundevents froma source j and depends on m^MC_t and the nuisance param- eters λ. Thedependence of thebackground processes onm^MC_t is introducednotonlybythecontributionoftW andsemileptonict¯t events,butalsobythechoiceofm^kin_t inthekinematicreconstruc- tion.Equation (2),whichrelatesthevarious

σ

_t⁽_¯t^μk) (andhencethe parton-leveldifferentialcrosssection)todistributionsoffinal-state observables, embeds the detector response and its parametrized

dependenceon thesystematicuncertainties. Therefore,themaxi- mizationofthelikelihoodfunctionprovidesresults for

σ

_t⁽_¯t^μk)that are automatically unfolded to the partonlevel.This method (de- scribed,e.g. inRef. [48])isalsoreferredtoasmaximum-likelihood unfolding and, unlikeother unfoldingtechniques,allows the nui- sance parameters to be constrained simultaneously withthe dif- ferential cross section. The unfolding problem was found to be well-conditioned, andtherefore no regularization is needed. The expected signal andbackground distributions contributing to the fitaremodelledwithtemplatesconstructedusingsimulatedsam- ples.

Selectedeventsare categorizedaccordingto thenumberofb- tagged jets, as events with 1 b-tagged jet, 2 b-tagged jets, or a different number of b-tagged jets (zero or more than two). The effectofthesystematicuncertainties onthe normalizationofthe differentsignalsineachofthesecategoriesisparametrizedusing multinomial probabilities. Inparticular, based onthe t¯t topology, the number of events with one (S^k_1b), two (S^k_2b), or a different number of b-tagged jets(S^k_other) in each bin ofm_t¯t is expressed as:

S^k_1b

=

L

σ

_t⁽_¯t^μk)A^k_sel

_sel^k 2

_b^k

(

1

−

^Ck_b

_b^k

),

(3) S^k_2b

=

L

σ

_t⁽_¯t^μk)A^k_sel

_sel^k C_b^k

(

^k_b

)

²

,

(4) S^k_other

=

L

σ

_t⁽_¯t^μk)A^k_sel

_sel^k

^

1

−

2

_b^k

(

1

−

C_b^k

_b^k

) −

C^k_b

(

_b^k

)

²



.

(5)

Here, L ^{is the integrated} luminosity, A^k_sel is the acceptance of the event selection in the m_t¯t bin k, and

^k_sel represents the ef- ficiency for an eventin the visible phase space to pass the full eventselection. The acceptance A^k_sel is definedas thefraction of t¯t eventsin thebink that,atthegenerator (particle)level,enter thevisiblephasespacedescribed inSection3,while _sel^k includes experimental selection criteria, e.g. isolation and trigger require- ments. Furthermore,

^k_b represents the b tagging probability and the parameter C^k_b accounts for any residual correlation between thetaggingoftwob jetsinat¯t event.Thequantities A^k_sel, _sel^k , _b^k, and C_b^k are determined from thesignal simulation and, although they are not free parameters of the fit, they vary according to theparameters λandm^MC_t .Ineachcategory,theremainingeffects ofthe systematicuncertainties on signalprocessesare treatedas shape uncertainties. Thequantities s^k_i inEq. (2) arethen derived fromthesignalshapeandnormalizationinthecorrespondingcat- egory.Inthisway,apreciseparametrizationofthedependenceof signalnormalizationsonthenuisanceparametersandm^MC_t isob- tained. In fact,the parameters inEqs. (3)–(5) are lesssubject to statisticalfluctuationsthanthes^k_i.

Inordertoconstraineachindividual

σ

_t⁽_t¯^μk),eventswithatleast twojetsarefurtherdivided intosubcategories ofm^reco_t¯t ,usingthe samebinningasform_t¯t (Table1).Thechoiceoftheinputdistribu- tions tothefitinthedifferenteventcategoriesissummarizedin Table2.Thetotalnumberofeventsischosenasinputtothefitfor allsubcategorieswithzeroormorethantwob-taggedjets,where thecontributionofthebackgroundprocessesisthelargest,inor- der to mitigatethe sensitivity of the measurement to the shape of the distributions ofbackground processes. The same choice is madeforthesubcategoriescorrespondingtothelastbininm^reco_t¯t , where the statistical uncertainty in both data and simulation is large,andforeventswithlessthantwojets, wherethekinematic reconstruction cannot be performed. In the remaining subcate- gories withone b-taggedjet,the minimuminvariant massfound whencombiningthereconstructedb jet andalepton,referred to as the m^min_b distribution, is fitted. This distribution provides the

Table 2

Inputdistributionstothefitinthedifferenteventcate- gories.Thenumberofjets,thenumberofb-taggedjets, thenumberofevents,andthe pTofthesoftestjetare denotedwith Njets,Nb, Nevents,and“jet p^min_T ”,respec- tively,whilethecategorycorrespondingtothebink in m^reco_t¯t isindicatedwith“m^reco_t¯t k”.

Nb=^{1 N}b=^{2 Other N}b

Njets<2 Nevents n.a. Nevents

m^reco_t¯t 1 m^min_b jet p^min_T Nevents m^reco_t¯t 2 m^min_b jet p^min_T Nevents m^reco_t¯t 3 m^min_b jet p^min_T Nevents m^reco_t¯t 4 Nevents Nevents Nevents

sensitivityto constrain m^MC_t [49]. In the remaining subcategories with two b-tagged jets, the pT spectrum of the softest selected jet in the event is used to constrain jet energy scale uncertain- tiesat smallvaluesof p_T, thekinematic rangewhere systematic uncertainties arethe largest.The distributionsused inthe fitare comparedtothedataafterthefitinthesupplementalmaterial.

The efficiencies of the kinematic reconstruction in data and simulationhavebeeninvestigatedinRef. [33] andtheywerefound todifferby0.2%.Therefore,theefficiencyinthesimulationiscor- rectedtomatchtheoneindata.Anuncertaintyof0.2%isassigned toeachbinofm_t¯t independently.Thesameuncertaintyisalsoas- signedto t¯t eventswithoneortwo b-taggedjets,independently.

For t¯t events with zero or more than two b-tagged jets, where the combinatorialbackground is larger, an uncertainty of0.5% is conservativelyassigned.Theseuncertaintiesaretreatedasuncorre- latedtoaccountforpossibledifferencesbetweenthedifferentm_t¯t binsandcategoriesofb-taggedjet multiplicity.Similarly,anaddi- tionaluncertaintyof1%isassignedtothesumofthebackground processes,independentlyforeach binofm^reco_t¯t , inordertoreduce thecorrelationbetweenthesignalandthebackgroundtemplates.

Theimpact oftheseuncertainties on thefinal resultsis foundto besmallcomparedtothetotaluncertainty.

The dependence of the signal shapes, ofthe parameters A^k_sel,

_sel^k ,

_b^k, and C^k_b, and of the background contributions on m^MC_t andonthenuisanceparameters λismodelledusingsecond-order polynomials [14]. In the fit, Gaussian priors are assumed for all thenuisanceparameters.Thenegativelog-likelihoodisthenmini- mized,usingthe Minuit program [50],withrespectto

σ

_t⁽_¯t^μk),m^MC_t , and λ. Finally, the fit uncertainties in the various

σ

_t⁽_¯t^μk) are de- terminedusing Minos [50].Additional extrapolationuncertainties, which reflect the impact of modelling uncertainties on A^k_sel, are estimatedwithouttakingintoaccounttheconstraintsobtainedin the visiblephase space [14]. Moreover, an additional uncertainty arising from the limited statistical precision of the simulation is estimatedusingMCpseudo-experiments [14],wheretemplatesare variedwithintheirstatisticaluncertaintiestakingintoaccountthe correlationsbetweenthenominaltemplatesandthetemplatescor- responding tothe systematic variations. The template dependen- cies are then rederivedand the fit to the datais repeated more thantenthousandtimes.Foreachparameterofinterest,theroot- mean-squareofthebestfitvaluesobtainedwiththisprocedureis takenasanadditionaluncertaintyandaddedinquadraturetothe totaluncertaintyfromthefit.

The measured

σ

_t⁽_¯t^μk) are shown in Fig. 2 and compared to fixed-ordertheoretical predictionsinthe MS schemeatNLO [51]

implemented for the purpose of this analysis in the mcfm v6.8 program [52,53].Inthe calculation,therenormalizationscale,

μ

r, andfactorizationscale,

μ

f,arebothsettomt.TheMS massofthe top quark evaluatedat thescale

μ

=^mt isdenoted withm_t(m_t).

Fig. 2. Measuredvaluesofσ_t^(μ_¯t^k)(markers)andtheiruncertainties(verticalerror bars)comparedtoNLOpredictionsintheMS schemeobtainedwithdifferentvalues ofmt(mt)(horizontallinesofdifferentstyles).Thevaluesofσ_t^(μ_¯t^k)areshownatthe representativescaleoftheprocessμk,definedasthecentre-of-gravityofbink in m_t¯t.Thefirstandlastbinscontainalleventswithm_t¯t<420GeV andm_t¯t>810GeV, respectively.

ThecalculationisinterfacedwiththeABMP16_5_nloPDFset [54], whichistheonlyavailablePDFsetwheremt istreatedintheMS scheme andwhere the correlations between the gluon PDF,

α

S, and m_t are taken into account. In the calculation, the value of

α

S at theZ boson mass,

α

S(mZ), isset to thevalue determined in the ABMP16_5_nlo fit, which in the central PDF corresponds to 0.1191 [54].Inorderto demonstratethesensitivitytothe top quark mass,predictionsford

σ

_t¯t/dm_t¯t obtainedwithdifferentval- uesofmt(mt)areshown.Furthermore,itisworthnotingthatthis methodprovidesacrosssectionresultwithsignificantlyimproved precision comparedtomeasurements thatperformunfoldingasa separatestep,e.g. astheonedescribedinRef. [33].

The dominantuncertainties inthe measured

σ

_t⁽_¯t^μk) are associ- ated withtheintegratedluminosity,theleptonidentificationeffi- ciencies,the jet energyscales and, atlarge m_t¯t,the modellingof thetopquark pT.Thetwolatteruncertaintiesaremarginallycon- strainedinthefit,whilethefirsttwoarenotconstrained.Further- more,thepost-fitvaluesofallnuisanceparametersarefoundtobe compatiblewiththeirpre-fitvalue,withinonestandarddeviation.

Thenumericalvaluesofthemeasured

σ

_t⁽_¯t^μk),theircorrelations,the impact of the various sources of uncertainty, and the pulls and constraints of the nuisance parameters related to the modelling uncertaintiescanbefoundinthesupplementalmaterial.

5. Extractionoftherunningofthetopquarkmass

The measured differential crosssection is used to extract the running of the top quark MS mass at NLO as a function of the scale

μ

=^mt¯t. The procedure is similar to the one used to ex- tract the running of the charm quark mass [12]. The value of mt(mt) is determined independently in each bin of m_t¯t from a

χ

² fit of fixed-order theoretical predictions at NLO to the mea- sured

σ

_t⁽_t¯^μk).The theoreticalpredictionsareobtainedasdescribed in Section 4 for Fig. 2. The

χ

² definition follows the one de- scribed inRef. [55],whichaccountsforasymmetriesinthe input uncertainties. Theextractedm_t(m_t)are then convertedtom_t(

μ

k) using the CRunDec v3.0 program [56], where

μ

k is the repre- sentative scale ofthe process ina given binof m_t¯t,asdescribed in Section 3. As relevant in a NLO calculation, the conversion is performed withone-loop precision, assuming five active flavours (n_f =^5)and

α

S(m_Z)=⁰.1191 consistentlywiththeusedPDFset.

Thisprocedure isequivalentto extractingdirectlymt(

μ

k) ineach bin.Furthermore,theresultdoesnotdependontheexactchoiceof

μ

k,providedthatitisrepresentativeofthephysicalenergyscaleof theprocess.Infact,achangein

μ

k wouldcorrespondtoachange inm_t(

μ

k) accordingto the RGE. The extracted valuesof m_t(

μ

k) andtheiruncertaintiescanbefoundinthesupplementalmaterial.

In orderto benefitfrom the cancellation ofcorrelated uncer- tainties in the measured

σ

_t⁽_¯t^μk), the ratios of the various mt(

μ

k) to m_t(

μ

2) are considered. In particular, the quantities r₁₂ = mt(

μ

1)/mt(

μ

2), r32=^mt(

μ

3)/mt(

μ

2), and r42=^mt(

μ

4)/mt(

μ

2) are extracted. With this approach the running of mt, i.e. the quantity predictedby the RGE (Eq. (1)),is accessed directly. The measurementatthescale

μ

2 ischosenasareferenceinorderto minimizethecorrelationbetweentheextractedratios.

Four different types of systematic uncertainty are considered for the ratios: the uncertainty in the various

σ

_t⁽_¯t^μk) in the visi- blephase space(referred toas fituncertainty), theextrapolation uncertainties,theuncertaintiesintheprotonPDFs,andtheuncer- taintyinthe valueof

α

S(mZ).Thefit uncertaintyincludesexper- imentalandmodellinguncertainties described inSection 3.Scale variations in the mcfm predictions are not performed, since the scale dependenceof mt is beinginvestigated at a fixed order in perturbationtheory.Infact,scalevariationsinthehard scattering crosssectionareconventionallyperformedasameansofestimat- ingtheeffectofmissinghigherordercorrectionsandaretherefore notapplicableinthiscontext.

Uncertainties in the proton PDFs affect the mcfm prediction andtherefore the extractedvalues of the various mt(

μ

_k). In or- derto estimate their impact, thecalculation is repeatedforeach eigenvectorofthePDFsetandthedifferencesintheextractedra- tiosare added inquadratureto yield thetotal PDF uncertainties.

In the ABMP16_5_nlo PDF set,

α

S(mZ) is determined simultane- ously with the PDFs, therefore its uncertainty is incorporated in thatofthePDFs.However,theuncertaintyin

α

S(mZ)alsoaffects the CRunDec conversionfrommt(mt)tomt(

μ

k).Thiseffectises- timatedindependentlyandisfoundtobenegligible.

Theimpactofextrapolationuncertaintiesisestimatedbyvary- ingthemeasured

σ

_t⁽_¯t^μk)withintheirextrapolationuncertainty,sep- arately for each source and simultaneously in the different bins in m_t¯t, taking the correlations into account. The various contri- butions are added in quadrature to yield the total extrapolation uncertainty.

Thecorrelationsbetweentheextractedmassesarisingfromthe fit uncertaintyare estimated usingMC pseudo-experiments, tak- ing the correlations between the measured

σ

_t⁽_¯t^μk) asinputs. The uncertaintiesarethenpropagated totheratiosusinglinearuncer- taintypropagation,takingtheestimatedcorrelationsintoaccount.

Thenumericalvaluesoftheratiosaredeterminedtobe:

r₁₂

=

¹

.

030

±

⁰

.

018

(

fit

)

⁺₋⁰₀^._.⁰⁰³₀₀₆

(

PDF+

α

S

)

⁺₋⁰₀^._.⁰⁰³₀₀₂

(

extr

),

r₃₂

=

⁰

.

982

±

⁰

.

025

(

fit

)

⁺₋⁰₀^._.⁰⁰⁶₀₀₅

(

PDF+

α

S

) ±

⁰

.

004

(

extr

),

r₄₂

=

⁰

.

904

±

⁰

.

050

(

fit

)

⁺₋⁰₀^._.⁰¹⁹₀₁₇

(

PDF+

α

S

)

⁺₋⁰₀^._.⁰¹⁷₀₁₃

(

extr

).

Here,thefituncertainty(fit),thecombinationofPDFand

α

S un- certainty (PDF+

α

S), and the extrapolation uncertainty (extr) are given. Themost relevantsources ofexperimental uncertainty are theintegratedluminosity,theleptonidentificationefficiencies,and thejet energyscale and resolution.Among modellinguncertain- tiesrelated to the powheg+pythia 8 simulation of the t¯t signal, thelargestcontributionsoriginatefromthescalevariations inthe partonshower,theuncertaintyintheshapeofthepT spectrumof thetopquark,andthematchingscalebetweenthematrixelement andthe partonshower. The statisticaluncertainties are found to benegligible.Thecorrelationsbetweentheratiosarisingfromthe fituncertaintyareinvestigatedusinga pseudo-experimentproce- durewhichconsistsinrepeatingtheextractionoftheratiosusing

Fig. 3. Extracted running of the top quark mass mt(μ)/mt(μref) compared to the RGE prediction at one-loop precision, with nf =5, evolved from the ini- tial scaleμ0=μref=⁴⁷⁶GeV (upper).The result iscompared tothe value of m^incl_t (mt)/mt(μref),wherem^incl_t (mt)isthevalueofmt(mt)extractedfromthein- clusivecrosssectionmeasuredinRef. [14],whichisbasedonthesamedataset.

Theuncertaintyinm^incl_t (mt)isevolvedfromtheinitialscaleμ0=^minclt (mt),which correspondstoabout163 GeV,usingthesameRGEprediction(lower).

pseudo-measurementsof

σ

_t⁽_t¯^μk),generatedaccordingtothecorre- spondingfittedvalues,uncertainties,andcorrelations.With

ρ

ikbe- ingthecorrelationbetweenr_i2 andr_k2,theresultsare

ρ

13=^13%,

ρ

14= −^45%,and

ρ

34=^11%.

Theextractedratiosm_t(

μ

k)/m_t(

μ

2)areshowninFig.3(upper) togetherwiththeRGEprediction(Eq. (1))atone-loopprecision.In the figure,the referencescale

μ

2 is indicated with

μ

ref, andthe RGE evolutioniscalculatedfromtheinitialscale

μ

0=

μ

ref.Good agreementbetweentheextractedrunningandtheRGEprediction isobserved.

For comparison, the MS mass of the top quark is also ex- tracted from the inclusive cross section measured in Ref. [14], using Hathor 2.0 [57] predictions at NLO interfaced with the ABMP16_5_nlo PDF set, and is denoted with m^incl_t (m_t). Fig. 3 (lower) comparestheextractedratiosmt(

μ

k)/mt(

μ

2)tothevalue ofm^incl_t (mt)/mt(

μ

2).The uncertaintyinm^incl_t (mt) includesfit,ex- trapolation,andPDFuncertainties,andisevolvedtohigherscales, while thevalue ofmt(

μ

2) inthe ratiom^incl_t (mt)/mt(

μ

2) istaken withoutuncertainty.Here,theRGEevolutioniscalculatedfromthe initial scale

μ

0=^minclt (mt),which corresponds toabout163 GeV.

Theextractedvalue ofm^incl_t (m_t)andits uncertaintycanbefound inthesupplementalmaterial.

Finally,theextractedrunningisparametrizedwiththefunction

f

(

x

, μ ) =

^{x [r}

( μ ) −

^1]

+

¹

,

(6) where r(

μ

)=^mt(

μ

)/mt(

μ

2) corresponds to the RGE prediction shown in Fig. 3 (upper). In particular, f(x,

μ

) corresponds to

r(

μ

)forx=^{1 andto1,i.e. norunning,forx}=^{0.Thebestfitvalue} forx,denotedwithx,ˆ isdetermined viaa

χ

² fitto theextracted ratiostakingthecorrelations

ρ

ikintoaccount,andisfoundtobe

ˆ

x

=

2

.

05

±

0

.

61

(

fit

)

⁺₋⁰₀^._.³¹₅₅

(

PDF

+ α

S

)

⁺₋⁰₀^._.²⁴₄₉

(

extr

).

The result shows agreement between the extracted running and theRGEpredictionatone-loopprecisionwithin1.1standarddevi- ationsintheGaussianapproximationandexcludestheno-running hypothesisatabove95%confidencelevel(2.1standarddeviations) inthesameapproximation.

6. Summary

InthisLetter,thefirstexperimentalinvestigationoftherunning ofthetop quark mass,mt, ispresented.Therunning isextracted from a measurement of the differential top quark-antiquark (t¯t) cross section as a function of the invariant mass of the t¯t sys- tem,m_t¯t.Thedifferential t¯t crosssection,d

σ

_t¯t/dm_t¯t,isdetermined at the parton level using a maximum-likelihood fit to distribu- tions of final-state observables, using t¯t candidate events in the e^±μ^{∓ channel. This technique allows the nuisanceparameters to} beconstrainedsimultaneouslywiththedifferentialcrosssectionin thevisiblephasespaceandthereforeprovidesresultswithsignifi- cantlyimprovedprecisioncomparedtoconventionalproceduresin whichtheunfoldingisperformedasaseparate step.The analysis isperformedusingproton-protoncollisiondataatacentre-of-mass energyof13 TeV recordedbytheCMSdetectorattheCERNLHCin 2016,correspondingtoanintegratedluminosityof35.9 fb⁻¹.

The runningmass m_t(

μ

), asdefined inthe modified minimal subtraction(MS)renormalizationscheme,isextractedatone-loop precision asa function ofm_t¯t bycomparing fixed-order theoreti- calpredictionsatnext-to-leadingordertothemeasuredd

σ

_t¯t/dm_t¯t. The extracted running of mt is found to be in agreement with the predictionof the corresponding renormalization group equa- tion,within1.1standarddeviations,andtheno-runninghypothesis is excluded at above 95% confidence level. The running of m_t is probeduptoascaleoftheorderof1 TeV.

Acknowledgements

WecongratulateourcolleaguesintheCERNacceleratordepart- ments for the excellent performance of the LHC and thank the technicalandadministrativestaffs atCERN andatother CMSin- stitutes for their contributions to the success of the CMS effort.

Inaddition,wegratefullyacknowledgethecomputingcentresand personneloftheWorldwideLHCComputingGridfordeliveringso effectivelythe computinginfrastructureessential to ouranalyses.

Finally, we acknowledge the enduring support for the construc- tionandoperation oftheLHCandthe CMSdetectorprovidedby thefollowingfunding agencies:BMBWFandFWF(Austria); FNRS andFWO (Belgium); CNPq,CAPES, FAPERJ, FAPERGS, and FAPESP (Brazil); MES (Bulgaria); CERN; CAS, MOST, and NSFC (China);

COLCIENCIAS (Colombia); MSESand CSF (Croatia); RPF (Cyprus);

SENESCYT (Ecuador); MoER, ERC IUT, PUT and ERDF (Estonia);

AcademyofFinland,MEC,andHIP(Finland);CEAandCNRS/IN2P3 (France); BMBF, DFG, and HGF (Germany); GSRT (Greece); NK- FIA (Hungary); DAE and DST (India); IPM (Iran); SFI (Ireland);

INFN(Italy);MSIPandNRF(RepublicofKorea);MES(Latvia);LAS (Lithuania);MOEandUM(Malaysia);BUAP,CINVESTAV,CONACYT, LNS,SEP,andUASLP-FAI(Mexico);MOS(Montenegro);MBIE(New Zealand); PAEC (Pakistan); MSHE and NSC (Poland); FCT (Portu- gal);JINR(Dubna);MON, ROSATOM,RAS,RFBR,andNRCKI(Rus- sia);MESTD(Serbia);SEIDI,CPAN,PCTI,andFEDER(Spain);MoSTR (Sri Lanka); Swiss Funding Agencies (Switzerland); MST (Taipei);

ThEPCenter,IPST,STAR, andNSTDA(Thailand);TUBITAKandTAEK (Turkey);NASU andSFFR (Ukraine);STFC(United Kingdom);DOE andNSF(USA).

Individuals have received support from the Marie-Curie pro- gramme and the European Research Council and Horizon 2020 Grant, contract Nos. 675440, 752730, and 765710 (European Union); the Leventis Foundation; the Alfred P. Sloan Founda- tion; the Alexander von Humboldt Foundation; the BelgianFed- eral Science Policy Office; the Fonds pour la Formation à la Recherche dans l’Industrie et dans l’Agriculture (FRIA-Belgium);

the Agentschap voor Innovatie door Wetenschap en Technolo- gie (IWT-Belgium); the F.R.S.-FNRS and FWO (Belgium) under the “Excellence of Science – EOS” – be.h project n. 30820817;

the Beijing Municipal Science & Technology Commission, No.

Z181100004218003; The Ministryof Education,Youth andSports (MEYS) of the Czech Republic; the Lendület (“Momentum”) Pro- gramme and the János Bolyai Research Scholarship of the Hun- garian Academy of Sciences, the New National Excellence Pro- gram ÚNKP, the NKFIA research grants 123842, 123959, 124845, 124850, 125105, 128713, 128786, and 129058 (Hungary); the Council of Science and Industrial Research, India; the HOMING PLUSprogrammeoftheFoundation forPolishScience,cofinanced from European Union, Regional Development Fund, the Mobility Plus programme of the Ministry of Science and Higher Educa- tion, the National Science Center (Poland), contracts Harmonia 2014/14/M/ST2/00428,Opus2014/13/B/ST2/02543,2014/15/B/ST2/

03998,and2015/19/B/ST2/02861,Sonata-bis2012/07/E/ST2/01406;

the National Priorities Research Program by Qatar National Re- search Fund; the Ministry of Science and Education, grant no.

3.2989.2017 (Russia); thePrograma Estatal de Fomento de la In- vestigación Científica y Técnica de Excelencia María de Maeztu, grantMDM-2015-0509andtheProgramaSeveroOchoadelPrinci- pado de Asturias;the ThalisandAristeiaprogrammes cofinanced byEU-ESFandtheGreekNSRF;theRachadapisekSompotFundfor Postdoctoral Fellowship, Chulalongkorn University and the Chu- lalongkorn Academic into Its 2nd Century Project Advancement Project (Thailand);theNvidiaCorporation;TheWelchFoundation, contractC-1845;andtheWestonHavensFoundation(USA).

Appendix A. Supplementarymaterial

Supplementarymaterialrelatedtothisarticlecanbefoundon- lineathttps://doi.org/10.1016/j.physletb.2020.135263.

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