K− over K+ multiplicity ratio for kaons produced in DIS with a large fraction of the virtual-photon energy

Contents lists available atScienceDirect

Physics

Letters

B

www.elsevier.com/locate/physletb

K

−

over

K

+

multiplicity

ratio

for

kaons

produced

in

DIS

with

a

large

fraction

of

the

virtual-photon

energy

.

The

COMPASS

Collaboration

a

r

t

i

c

l

e

i

n

f

o

a

b

s

t

r

a

c

t

Articlehistory: Received26April2018

Receivedinrevisedform4September2018 Accepted26September2018

Availableonline9October2018 Editor: M.Doser

The K−overK+ multiplicityratioismeasuredindeep-inelasticscattering,forthefirsttimeforkaons carrying a large fraction z of the virtual-photon energy. The data were obtained by the COMPASS collaboration usinga160GeV muon beamandan isoscalar6_{LiDtarget.The regimeofdeep-inelastic}

scatteringisensuredbyrequiringQ2_>1(GeV/c)2_{forthephotonvirtualityand}

_W

_{>5 GeV/c}2_forthe

invariantmassoftheproducedhadronicsystem.Kaonsareidentifiedinthemomentumrangefrom12 GeV/c to 40GeV/c,therebyrestrictingtherangeinBjorken-x to0.01<x <0.40.The

z-dependence

of themultiplicityratioisstudiedfor

z

>0.75.Forverylargevaluesof

z, i.e. z

>0.8,weobservethekaon multiplicityratiotofallbelowthelowerlimitsexpectedfromcalculationsbasedonleadingand next-to-leadingorderperturbativequantumchromodynamics.Also,thekaonmultiplicityratioshowsastrong dependence onthe missingmassofthesingle-kaonproductionprocess.Thissuggeststhatwithinthe perturbativequantumchromodynamicsformalismanadditionalcorrectionmayberequired,whichtakes intoaccountthephasespaceavailableforhadronisation.

©2018TheAuthor(s).PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense (http://creativecommons.org/licenses/by/4.0/).FundedbySCOAP3_.

1. Introduction

Quark fragmentation into hadrons is a process of fundamen-talnature.Inperturbativequantum chromodynamics(pQCD),this processis effectivelydescribed by non-perturbativeobjects called fragmentationfunctions(FFs).Whilethesefunctionspresently can-not be predicted by theory,their scale evolution is described by the DGLAPequations [1]. Inleading order(LO) pQCD, the FF Dh q

represents a probability density,which describes the scaled mo-mentum distribution ofa hadron type h that is produced inthe fragmentationofaquarkwithflavour q.

ThecleanestwaytoaccessFFsistostudyhadronproductionin single-inclusiveannihilation,e+

+

e−

→

h

+

X ,wherethe remain-ing final state X is not analysed. These studies have two disad-vantages:i

)

thatonlyinformationaboutDh

q

+

Dhq_¯ isaccessible,and ii

)

withoutinvokingmodel-dependentalgorithmsforquark-flavour taggingonlylimitedflavour separationispossible.Incontrast,the analysisofsemi-inclusivemeasurements ofdeep-inelasticlepton– nucleonscattering(SIDIS)isadvantageousinthat qandq can

¯

be accessedseparatelyandfullflavourseparationispossiblein prin-ciple.Here, thedisadvantageisthat inthepQCD descriptionofa SIDISmeasurementFFsappearconvolutedwithpartondistribution functions(PDFs).

Recently, COMPASS reported results on charged-hadron, pion andkaonmultiplicitiesobtainedoverawidekinematicrange [2,3]. Theseresultsprovideimportantinputforphenomenological

analy-sesofFFs.Thepionmultiplicitieswerefoundtobewelldescribed bothinleading-order(LO) andnext-to-leadingorder(NLO)pQCD, while thiswasnot thecaseforkaon multiplicities.Theregion of large z appearstobeparticularlyproblematic forkaons,asitwas also observed insubsequent analyses [4] ofthe COMPASS multi-plicities.Here, z denotesthefractionofthevirtual-photonenergy carriedbytheproducedhadroninthetargetrestframe.

InthisLetter,wepresentresultsontheK− overK+ multiplic-ityratiointhelarge-z region,i.e. for z

>

0

.

75.Insteadofstudying multiplicities for K− and K+ separately, their ratio RK is

anal-ysed as inthiscase mostexperimental systematiceffects cancel. Similarly, the impactoftheoretical uncertainties, e.g. scale uncer-tainties, is largely reducedin the ratio.Also, while pQCD cannot predictvaluesofmultiplicities,limitsforcertainmultiplicityratios can be predicted.The Letter isorganisedasfollows: inSection 2

various predictionsfor RK are discussed.Theexperimental set-up

and the data selection are described in Section 3. The analysis method is presented in Section 4, followed by the discussion of thesystematicuncertaintiesinSection5.Theresultsarepresented anddiscussedinSection6.

2. Theoreticalframeworkandmodelexpectations

Hadrons oftype hproducedinaSIDISmeasurementare com-monlycharacterisedbytheir relativeabundance.Thehadron mul-tiplicity Mh is definedas theratio ofthe SIDIScross section for

https://doi.org/10.1016/j.physletb.2018.09.052

0370-2693/©2018TheAuthor(s).PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense(http://creativecommons.org/licenses/by/4.0/).Fundedby SCOAP3_.

hadrontype h to thecross section foran inclusivemeasurement ofthedeep-inelasticscatteringprocess:

dMh

(

x

,

Q2

,

z

)

dz

=

d3

σ

h

₍

_x

_,

_Q2

_,

_z

_)/

_dxdQ2_dz

d2

_σ

DIS

₍

_x

_,

_Q2

_)/

_dxdQ2

.

(1)

Here, Q2 _{is the virtuality of the photon mediating the lepton–}

nucleonscatteringprocessandx denotestheBjorkenscaling vari-able. Within the standard factorisation approach of pQCD [5,6],

σ

DIS _{can be written as a sum over parton types, in which for}

a givenparton type a therespective PDF is convoluted withthe lepton–parton hard-scattering cross section. For

σ

h _{in the}

cur-rentfragmentationregion,thesumcontainsanadditional convolu-tionwiththefragmentationfunctionoftheproducedparton. The rathercomplicatedNLOexpressionsforthesecrosssectionscanbe founde.g. inRef. [6].Below,wewilluseonlypQCDLOexpressions for the cross section, while later for the presentation of results alsomultiplicity calculationsobtainedusingNLO expressions will beshown.ItisimportanttonotethatintheSIDISfactorisation ap-proachtheonlyingredients that dependonthe nucleontype are thenucleonPDFs,whilethe fragmentationfunctionsdepend nei-theronthenucleontypenoronx.IntheLOapproximationforthe multiplicity,thesumoverpartonspeciesa

=

q

,

q does

¯

notcontain convolutionsbutonlysimpleproductsofPDFs fa

(

x

,

Q2

)

,weighted bythe square oftheelectriccharge ea of thequark expressedin unitsofelementarycharge,andFFs Dh_a

(

z

,

Q2

)

:

dMh

(

x

,

Q2

,

z

)

dz

=



ae2a

_

fa

(

x

,

Q2

)

Dah

(

z

,

Q2

)

ae2afa

(

x

,

Q2

)

.

(2)

For a deuteron target, the charged-kaon multiplicity ratio in LO pQCDreadsasfollows: RK

(

x

,

Q2

,

z

)

=

dMK −

(

x

,

Q2

,

z

)/

dz dMK+

₍

_x

_,

_Q2

_,

_z

_)/

_dz

=

4

(

u

¯

+ ¯

d

)

Dfav

+ (

5u

+

5d

+ ¯

u

+ ¯

d

+

2

¯

s

)

Dunf

+

2sDstr 4

(

u

+

d

)

Dfav

+ (

5u

¯

+

5d

¯

+

u

+

d

+

2s

)

Dunf

+

2sD

¯

str

.

(3)

Here,u, u,

¯

d, d,

¯

s,s denote

¯

thePDFs intheproton for differ-ent quark flavours.Their dependences on x and Q2 are omitted forbrevity. Thesymbols Dfav, Dunf and Dstr denotefavoured,

un-favoured, andstrange-quark fragmentationfunctions respectively, whicharegivenbyDfav

=

DK

+ u

=

DK − ¯ u ,Dunf

=

DK + ¯ u

=

DK + d

=

DK + ¯ d

=

DK_s+ andtheircharge conjugate,and Dstr

=

DK

+

¯

s

=

D K−

s .Their

de-pendenceson z and Q2 areomitted.Accordingly, alsothe depen-denceof RK on x, Q2 andz are omitted.Presently, existingdata

donot allow oneto distinguish betweendifferentfunctions Dunf

fordifferent quark flavours.However, it is expected that Dunf is

small in the large-z region, and this expectation is indeed con-firmedin pQCD fits alreadyat moderatevalues of z,i.e. z

≈

0

.

5, seee.g.Refs. [7,8].Whenneglecting Dunf,Eq. (3) simplifiesto

RK

=

4

(

u

¯

+ ¯

d

)

Dfav

+

2sDstr 4

(

u

+

d

)

Dfav

+

2sD

¯

str

.

(4)

Itisexpectedthat Dstr

>

Dfav

>

0,andthereforethepositiveterms

sDstrand

¯

sDstrmaybeofsomeimportance.Still,inorderto

calcu-latealower limitforRK,thesetermscan beneglectedunderthe

assumptionthats

= ¯

s,whichleadsto

RK

>

¯

u

+ ¯

d

u

+

d

.

(5)

Theanalysisdescribedbelowisperformedusingtwobinsin x, i.e. x

<

0

.

05 with



x



=

0

.

03,



Q2



=

1

.

6

(

GeV

/

c

)

2 and x

>

0

.

05 with



x



=

0

.

094,



Q2



=

4

.

8

(

GeV

/

c

)

2.Wheneversufficient, only thefirstx-binisusedinthediscussion.

The evaluationofEq. (5) forx

=

0

.

03 and Q2

₌

₁

_.

₆

₍

_GeV

_/

_c

₎

2

yieldsalowerlimitof0

.

469

±

0

.

015 whenusingtheMSTW08LO PDFs [9]. In NLO the limit given by Eq. (5) receives corrections on the level of

∼

α

S

/

2

π

. Using the MMHT14 NLO PDF set [10], the ratio

(

u

¯

+ ¯

d

)/(

u

+

d

)

is 0

.

440

±

0

.

023, but according to our calculationthelowerlimitisabout15%lowerthanthislimit.1

Wenotethatbecauseofthelargeuncertaintiesofs

,

¯

s andDstr,

reasonableuncertaintiesarepresentlycalculableonlyforthelower limitsof RK, andnot for RK itself.Theseuncertaintiesamountto

about3% forLOandabout6% forNLOpredictions. Inboth cases the uncertainty of the

(

u

¯

+ ¯

d

)/(

u

+

d

)

ratio dominates, while in NLOalsouncertaintiesofthegluonPDFplaysomerole.Thechoice ofFFshasnegligibleimpactonLOorNLOcalculationsofthelower RKlimit.Theactualpredictionsfor RKbasedonDSS [7] atLO

ac-curacyandDEHSS17 [8] atNLOaccuracyarelargerthanthelower limits for RK, which is expected as in the above calculation of

lowerlimitsthestrange-quarkcontributiontokaonfragmentation was neglected. It was verified that when using morerecent PDF sets(e.g. NNPDF30atLOandNLOaccuracy [11]),theRKvalues

in-creasebyabout10%forallcasesthatwerediscussedabove.Hence

ourchoiceoftheMSTW08LOandMMHT14NLO PDFssetsleads

toaratherconservativeestimationofthelowerlimitonRK.

IntheLEPTOeventgenerator2_{[12] another factorisationansatz}

isused dMh

(

x

,

Q2

,

z

)

dz

=



aea2fa

(

x

,

Q2

)

Hah/N

(

x

,

z

,

Q2

)



ae2afa

(

x

,

Q2

)

.

(6) Here, Hh

a/N

(

x

,

z

,

Q2

)

describestheproductionofahadronhinthe hadronisation of astring that is formedby the struck quark and thetargetremnant.IncontrasttothepQCDapproach,this hadro-nisationfunctiondependsnotonlyonquarkandhadrontypesand on z but also on the type of the target nucleon and on x, see Ref. [14] formoredetails. Wenote thatinthisapproachalso the conservationoftheoverall quantumnumbersaswellas momen-tumconservationaretakenintoaccount,whichisnotthecasefor thepQCD approach.TheLEPTOpredictionforRK,about0.52, lies

abovetheLOlimitgivenbyEq. (5).However,forz

>

0

.

97 it under-shootsthislimit.Thisappearsplausibleasforz approachingunity K+ can be producedin theprocess

μ

p

→

μ

K+



0,whilea simi-larprocesstoproduceK− isforbiddenbecauseofbaryonnumber conservation.

In recent years, severaltheory developments were performed that can potentially impact the theory predictions forthe high-z region.InRef. [15] forexample,theauthorsstudiedtheimpactof threshold-logarithmresummationsinthehigh-z regionandfound a largeimpact.Inthe caseof

π

− production,thepredictedcross section can be larger by a factor of two. When considering the

1 _{Fromtheformalismgivenin[5],itfollowsthatintheNLOcross-section}

for-mulaforhadronproduction,foreachquarkflavourtherearesixadditionalterms besidestheqDh

q term.Thesetermsinclude convolutionintegralsof PDF,FFsand

theso-calledcoefficientfunctions.Wefoundthatfourconvolutionintegralscan ef-fectivelybeneglectedathighz,andonlytwothatarerelatedtoconvolutionsof C1

qqandC1qghaveanimportantimpactonthefinalresults.ThetermrelatedtoCqq1

alonewouldleadtoanincreaseofRKabovethelimitgivenbyEq. (5).Incontrast,

thetermrelatedtoC1

qg,althoughappearinginasymmetricforminnumeratorand

denominator,isnegative,sothatthelowerlimitof RKfallsbelowthatgivenby

Eq. (5).WenotethatDfavoritsconvolutionappearsalwaysinallrelevantterms.

Itschoicehenceappearstoberatherirrelevantforthefinalresult,asitlargely can-celsinthepredictedlowerlimitforRKatNLO.

Fig. 1. Acceptance-uncorrected distributions of selected events in the ( Q2_{, x) plane and in the (ν, z) plane.} lower limit for RK, the resummation corrections for K− and K+

are largely proportional to the PDF densities u

¯

+ ¯

d and u

+

d, respectively.Therefore,the RK predictionsincludingthese

resum-mationcorrectionswouldbeevenclosertotheexpectationsgiven byEq. (5) thantheNLOpredictionsshownbelowwithout includ-ingthesecorrections.An interestingworkrelatedtohadron-mass corrections[16] was originally criticisedin Ref. [17], butthe dis-cussion is ongoing [18]. The approach discussed inthis work al-lowsonetoobtainavalueofRKbelowthelimitsdiscussedabove.

However, thisapproach seemsto gobeyondthestandard factori-sationtheorem andcorrectionsto Dh_q are needed, whichdepend onthetype oftargetnucleonandproducedhadronh.Therewere alsoother developments,e.g. Refs. [19–21],whicharevery impor-tantforabetterunderstanding ofthe hadronisationprocess.Still, theyappeartonoteffectivelyimpactthepredictionsforRKinthe

high-z regionatCOMPASSkinematics. 3. Experimentalset-upanddataselection

The data were taken in 2006using a

μ

+ beam delivered by

the M2 beam line of the CERN SPS. The beam momentum was

160 GeV

/

c withaspreadof

±

5%.Thesolid-state6_LiDtargetis

con-sideredto bepurely isoscalar, neglecting the0.2% excessof neu-tronsoverprotonsduetothepresenceofadditionalmaterialinthe target(3_Heand7_{Li).Thetargetwaslongitudinallypolarisedbutin}

thepresentanalysisthedata areaveraged overthetarget polari-sation, whichleads toan effectively vanishing target polarisation ona levelofbetter than1%. TheCOMPASS two-stage spectrome-terhasapolarangleacceptanceof

±

180mrad,anditiscapable ofdetectingchargedparticleswithmomentaabove0.5GeV/c.The ring-imaging Cherenkovdetector (RICH) was used to identify pi-ons,kaons andprotons.Its radiatorvolumewas filledwithC4F10

leading toa thresholdforpion,kaonandprotonidentification of about3 GeV/c,9 GeV/c and 18GeV/c respectively. Efficient pion andkaonseparationispossiblewithhighpurityformomenta

be-tween 12 GeV/c and 40 GeV/c. Two trigger types were used in

theanalysis. The“inclusive”triggerwas basedona signalfroma combinationof hodoscope signalsfrom the scattered muon. The “semi-inclusive” trigger required an energy deposition in one of the hadron calorimeters.The experimental set-up isdescribed in moredetailinRef. [22].

Thedataselectioncriteriaarekeptsimilartothoseusedinthe recentlypublished analysis[3], wheneverpossible.Thekinematic domain Q2

>

1

(

GeV

/

c

)

2 andW

>

5 GeV

/

c2 isselected,thereby restricting the analysis to the region of deep inelastic scattering wherepQCDcanbeapplied.Forsmallvaluesof y,i.e. thefraction of theincoming muon energy carried by the virtual photon, the momentumresolutionisdegraded.Inordertoexcludethisregion, y is required to have a minimum value of 0.1. The aim of this

analysisistostudykaonproductioninSIDISforkaons carryinga largefractionz of thevirtual-photonenergy,henceitisrestricted to z

>

0

.

75.Using theabove givenmomentumrangeforefficient kaon identification together with the large-z requirement inthis analysisleadstoaneffectiveupperlimitfory of0.35.

The kaonmultiplicities MK_{(x, Q}2_{, z) aredetermined fromthe}

kaonyields NK_{normalisedbythenumberofDISevents,N}DIS_,and

dividedbytheacceptancecorrection AK

(

x

,

Q2

,

z

)

:

dMK

(

x

,

Q2

,

z

)

dz

=

1 NDIS

₍

_x

_,

_Q2

₎

dNK

(

x

,

Q2

,

z

)

dz 1 AK

₍

_x

_,

_Q2

_,

_z

₎

.

(7)

Note that in this work “semi-inclusive” triggers can be used be-causeabiasfreedeterminationofNDISisnotneeded,asthelatter cancelsinRK.

Alldatatakenin2006areusedintheanalysis;altogetherabout 64000chargedkaonsareavailableintheregionz

>

0

.

75.Examples ofacceptance-uncorrecteddistributionsofselectedeventsare pre-sented in Fig. 1in the (x, Q2) and (

ν

, z) planes.Here,

ν

is the energyofthevirtualphotoninthelaboratoryframe.

4. Analysismethod

The analysis is performed in two x-bins, below and above

x

=

0

.

05, as already mentioned in Section 2. In each x-bin, five bins are used in the reconstructed z variable(zrec) withthe bin

limits0.75,0.80,0.85,0.90,0.95,1.05.SincetheRICHperformance dependuponthemomentumoftheidentifiedkaon,wealsostudy RKinbinsofthisvariableusingthebinlimits12GeV/c,16GeV/c,

20 GeV/c, 25GeV/c, 30GeV/c, 35 GeV/c,40 GeV/c.Note that in thiswaythe

ν

dependenceofRKisstudiedimplicitlyandthatthe

resultsarealsogivenasafunctionof

ν

inthesekaon-momentum bins.

In order to determine the multiplicity ratio RK from the raw

yield ofK− andK+ mesons,severalcorrection factorshaveto be taken into account. First, the number of identified kaons is cor-rected for the RICH efficiencies. Based on studies of

φ

→

K+K− decays,wherethe

φ

mesonwasproducedinaDISprocess,the ef-ficiencyratioforthetwochargesisfoundtobe1

.

002

±

0

.

012.Such a simple “unfolding” procedure can be followed because a strict selection ofkaons is made, so that the probabilities of misiden-tification of pionandprotonaskaon canbe assumedto be zero (possibleremainingmisidentificationprobabilitiesarediscussedin Section5).

Theacceptancecorrectionfactors AK _{forthetwokaoncharges}

are determined using Monte Carlo simulations. In the previous COMPASSanalysis[3],asimpleunfoldingmethodwasusedto de-termine these factors. For a given kinematic bin in

(

x

,

y

,

z

)

, the acceptance was calculated as the ratio of the number of

recon-Fig. 2. Left:TheK−overK+acceptanceratiointhefirstx-bin,i.e. x<0.05,asafunctionofthereconstructedz variable,asobtainedfromaMonteCarlosimulation.Right: Thecharged-kaonmultiplicityratiointhefirstx-bin,asafunctionofthelowerlimitoftheRICHlikelihoodratioforkaonswithmomentabetween35GeV/c and40GeV/c. Thearrowmarksthevalueusedintheanalysis(seetextformoredetails).

structed events to that of generatedones. Fora given event, re-constructedvariableswereusedtocountreconstructedeventsand generatedvariablestocountgeneratedevents.Inordertoaccount forthestrong z-dependence ofthemultiplicity inthelarge-z re-gion,inthisanalysistheacceptanceisunfolded asinRef. [3] for x and Q2 butnotforz.Variousmethodsforz unfoldingwere in-vestigatedindetailedstudies,seeappendixAforan example.The resultspresentedinthisLetterareobtainedusingthesimplest ver-sionof z unfolding, i.e. unfoldingonly the dependenceof RK on zcorr.Here,zcorr denotesthereconstructedvalueofz inthe

exper-iment,correctedbytheaverage differencebetweenthegenerated andreconstructedvaluesofz,wherethelatteraredeterminedby MonteCarlosimulations. In the left panel of Fig.2, the K− over K+ acceptanceratioobtainedfromx and Q2 _{unfoldingisshown}

asa functionof thereconstructed z-variableinthe first x-bin. It appearstobeindependentofz withinstatisticaluncertaintiesand hasavalue of0

.

921

±

0

.

004 inthe firstx-binand0

.

969

±

0

.

010 inthesecond x-bin.

Thecontaminationby decayproductsofdiffractivelyproduced vector mesons is estimated using HEPGEN [23] and found to be negligible,seeFig. 2in[3].Only

φ

decaysaresimulatedtheresince heaviervector mesons have crosssections smaller by a factorof about10anddecaymostlyinmulti-bodychannels, whichresults inevensmallerprobabilitiestoproducekaonsatlargez.

Themeasuredcrosssectionshavetobecorrectedforradiative effects inorder to obtain

σ

DIS _and

_σ

h_{. Since, y}

_<

₀

_.

_{35 holds as}

explainedabove,thesizeofradiativecorrectionsisexpectedtobe small.Inanycase,

σ

DIS _cancelsinR

K andintheTERADcode[24]

usedinCOMPASS analyses the relativeradiative correction isthe sameforK+andK−,sothatitalsocancelsintheratio.

5. Systematicstudies

Thecharged-kaon multiplicity ratiosmeasured inthisanalysis are found to agree with the results of the previous analysis [3] inthe overlapregion ofthe z-ranges usedin thesetwo analyses

(

0

.

75

<

z

<

0

.

85

)

. Results derived from data that were obtained usingdifferenttriggersarefoundtoagreewithoneanotherwithin 2%.

ThemostimportantcorrectionfactoristheK−overK+ accep-tanceratio,whichforthefirstx-binis0

.

921

±

0

.

004,asobtained usingMonteCarlodata.TheCOMPASSspectrometerisdesignedto bealmost charge symmetric.In thecaseofpions,theacceptance ratioobtainedfromMonteCarlosimulationsis0

.

991

±

0

.

003,i.e. veryclosetounity. Incontrast,theacceptanceratioofkaons ob-tainedfromMonteCarloisfoundto besignificantly belowunity. Thisdifference betweenK− andK+ yields iscaused bythe

non-negligible thickness of the COMPASS target, which amounts to

about50% of a hadron interaction length, combinedwith a con-siderably larger absorptioncross section forinteractions of nega-tive kaons compared to positive ones, see e.g. the results on the K±-deuteroncrosssection inRef. [25].Dependingonthe longitu-dinalpositionoftheprimaryinteractionpoint Zvtx,theproduced

kaonstraverseavaryingthicknessofthematerialcontainedinthe 120cmlongtarget.Asaresult,morenegativethanpositivekaons areabsorbedwhenthe interactiontookplace atthebeginning of thetarget ascompared toaninteraction attheendofthetarget. Itisverifiedthat oncetheacceptancecorrectionwasapplied,the obtainedRK ratioisflatasafunctionof Zvtx.FortheK−overK+

acceptanceratioa2% systematicuncertaintyisused;thisvalueis dominated by possibletrigger-dependent variations of the multi-plicitiesmentionedinthepreviousparagraph.

ThestabilityofRKistestedondatausingseveralvariablesthat

aredefinedinthespectrometercoordinatesystem.Themost sen-sitive one is the azimuthal angle

φ

of the produced kaon. The direction

φ

=

0 lies in the bending plane of the dipole magnets andpoints towardsthe side,to whichpositive particlesare bent. Correspondingly,the direction

φ

=

π

/

2 pointstowardsthetop of the spectrometer. In certain cases the charged-kaon multiplicity ratio is found to vary by up to 25%, withparticularly small val-uesclosetoa peakat

φ

=

0.Thisobservationisaccountedforby a systematic uncertainty that is takenas the difference between themultiplicity ratiomeasured overthefull

φ

-rangeandtheone measured for

|φ|

>

0

.

5. Typically,the relativeuncertainty related tothis

φ

-dependencerangesbetween3%and11%,whichmakesit thedominantsystematicuncertainty. Notethatthe valuesofthis systematic uncertainty fordifferent bins in z are strongly corre-lated,withacorrelationcoefficientofabout0.8.

FurthersystematicuncertaintiesmayarisefromtheRICH iden-tification procedure.The K− over K+ efficiencyratio is expected to be closeto unitysince theRICH detector issituatedbehind a dipolemagnetofrelativelyweakbendingpower.Additionalstudies wereperformedondataconcerningmisidentificationprobabilities ofpionsandprotonsbeingidentifiedaskaonsby varyingthe ra-tioofthekaonlikelihood,whichisthelargestofalllikelihoodsin the selected sample, to the next-to-largest likelihood hypothesis, LK

/

L2nd.Thebehaviour of RK asafunction ofthelowerlimit for LK

/

L2nd is shownin therightpanel ofFig.2forkaon candidates

with momenta between 35 GeV/c and 40 GeV/c. The constraint

LK

/

L2nd

>

1

.

5 is used in the present analysis. From these

stud-ies,thesystematicuncertaintyoftheRICHunfoldingprocedureof about3%. It corresponds to the difference in RK calculated from

thefinal sample andtheone,inwhich anon-zero

π

contamina-tionisdetected.

Table 1

ExtractedvaluesofRK,binlimitsofz(zmin, zmax),andtheaveragesvaluesofx,Q2,zrecandzcorrinfirst

(upperpart)andsecond(lowerpart)x-bin. Bin x Q2_(GeV/c)2 _z

min zmax zrec zcorr RK± δRK,stat.± δRK,syst.

1 0.030 1.7 0.75 0.80 0.774 0.771 0.401±0.007±0.019 2 0.030 1.6 0.80 0.85 0.824 0.817 0.350±0.008±0.018 3 0.031 1.6 0.85 0.90 0.873 0.860 0.287±0.008±0.015 4 0.031 1.6 0.90 0.95 0.923 0.900 0.228±0.009±0.015 5 0.032 1.5 0.95 1.05 0.982 0.934 0.150±0.009±0.017 1 0.094 5.1 0.75 0.80 0.774 0.771 0.235±0.007±0.009 2 0.094 4.8 0.80 0.85 0.824 0.817 0.204±0.007±0.011 3 0.093 4.6 0.85 0.90 0.873 0.860 0.177±0.008±0.010 4 0.093 4.4 0.90 0.95 0.923 0.900 0.136±0.008±0.016 5 0.093 4.2 0.95 1.05 0.982 0.934 0.090±0.008±0.010

As the COMPASS muon beam is (naturally) polarised with an

average polarisationof

−

0

.

80

±

0

.

04, a spin-dependent contribu-tiontothetotallepton–nucleoncrosssection cannotbeneglected apriori.Thiscontributionisproportionaltosin

φ

handexpectedto

be smallerthan the spin-independent one, whichis proportional to cos

φ

h andcos 2

φ

h [26]. Here,

φ

h denotes the azimuthal angle

between the lepton-scattering plane and the hadron-production plane inthecentre-of-mass frameofvirtual photon andnucleon. Studies performed for previous COMPASS measurements [2,3] showthattheseeffectscanbeneglectedwhenusing

φ

h-integrated

multiplicities,asitisdoneinthisanalysis.

Altogether, the total relative systematic uncertainty on RK is

found to range between5% and 12% depending upon the z-bin.

The systematicuncertainties in different z-binsare highly corre-lated, i.e. thecorrelation coefficient is estimatedtovary between 0.7and0.8.

6. Resultsanddiscussion

In Table 1, the results on the charged-kaon multiplicity ra-tio RK are presented in bins of the reconstructed z variable for

the two x-bins. The measured z-dependence of RK can be

fit-ted in both x-bins by simple functional forms, e.g.

∝ (

1

−

z

)

β_,

β

=

0

.

71

±

0

.

03.Dividingineveryz-binthevalueoftheratio mea-suredinthefirstx-binbytheonemeasuredinthesecondx-bin,a “doubleratio”DK

=

RK

(

x

<

0

.

05

)/

RK

(

x

>

0

.

05

)

isformedthat

ap-pearsto be constant over all themeasured z-range witha value DK

=

1

.

68

±

0

.

04stat.

±

0

.

06syst.. It is interesting to note that the

measuredvalueagreeswithinuncertaintieswithDKcalculated

us-ingtheLOMSTW08LPDFset,i.e. 1

.

56

±

0

.

07.InFig.3,RKisshown

asafunctionof zcorr forthetwo x-bins,aswell asDK in the

in-setofthefigure.AsbothdataandLOpQCDcalculationexhibitthe samez-dependence whencomparing thecharged-kaon multiplic-ityratiosin thetwo x-bins,inwhat followswe concentrateonly onthe firstx-bin, i.e. x

<

0

.

05.Still,theconclusions presentedin theremainingpartoftheLetterarevalidforbothx-bins.

InFig.4,thepresentresultson RK inthefirst x-binare

com-paredwiththeexpectations fromLOandNLO pQCD calculations andwiththe predictionsobtainedusingtheLEPTOevent genera-tor, whichwere all discussed inSection 2. Forcompleteness, we notethatinthesecond x-binthetypical RKpredictionsareabout

1.5–1.6 times smaller than in the first x-bin. It is observed that withincreasing z thevaluesof RK are increasinglyundershooting

the expectations from LO andNLO calculations. The discrepancy

between the COMPASS results and the NLO predictions reaches

a factor of about2.5 atthe largest value of z. As the difference betweenthelowerlimit inLOandtheNLODEHSSprediction ob-tainedundertheassumptionDstr

=

0 isneverlargerthan20%,itis

veryunlikelythatanypredictionobtainedatNNLOwouldbeable toaccountforsuchalargediscrepancy.

Fig. 3. ResultsonRKasafunctionofzcorrforthetwox-bins.Theinsertshowsthe

doubleratioDKthatistheratioofRKinthefirstx-binover RK inthesecond x-bin.Statisticaluncertaintiesareshownbyerrorbars,systematicuncertaintiesby theshadedbandsatthebottom.

Fig. 4. ComparisonofRKinthefirstx-binwithpredictionsdiscussedinSection2.

Thesystematicuncertaintiesofthedatapointsareindicatedbytheshadedbandat thebottomofthefigure.Theshadedbandsaroundthe(N)LOlowerlimitsindicate theiruncertainties.

As alreadymentioned in Sect.2, thepresented pQCD calcula-tions rely on the factorisation ansatz d3

σ

h

₍

_x

_,

_Q2

_,

_z

_)/

_dxdQ2_dz

_∝



aea2fa

(

x

,

Q2

)

Dah

(

z

,

Q2

)

. If this ansatz would not be applicable atCOMPASSenergies forlargevaluesofz,itmaybeincapableto describethebehaviour ofkaonmultiplicitiesinthiskinematic re-gion.ThispQCDansatzdoesnotincludehigher-twistterms,which areproportionaltopowers1

/

Q2,sothattherespectivecorrection should be smallerby a factorofaboutthree inthe second x-bin

compared to the first x-bin. However, the discrepancy between

Fig. 5. TheK− overK+ multiplicityratioasafunctionofνinbinsofz,shownforthefirstbininx.Thesystematicuncertaintiesofthedatapointsareindicatedbythe shadedbandatthebottomofeachpanel.Theshadedbandsaroundthe(N)LOlowerlimitsindicatetheiruncertainties.

bethesameinthe twox-bins withinexperimental uncertainties. The observed discrepancy cannot be explained by the threshold resummationsfromRef. [15],asdiscussedinSection2.Theusage ofDSSfragmentationfunctions[7] intheLOansatz presentedin Ref. [16] leadstoadecreaseofthe RKpredictionbyabout25%in

thelast z bin.It is thus not enoughto account forthe observed discrepancy.However, largerchangescould beobtainedifFFs de-creaseto zerofasterthanexpectedintheDSS parametrisation.It isworth notingthat intheLEPTOeventgeneratoradifferent fac-torisationapproachisused,whichisbasedonstringhadronisation. However,itdoesnotdescribethedataathighz,inspiteofits con-siderably higher flexibility in comparisonto the pQCD approach. Perhapsa special tuning ofcertain string fragmentation parame-ters,for examplethose governing low-mass string hadronisation, wouldleadtoabetterdescriptionofthedata.

Inthe analysis we assume that there is no contamination by decayproducts ofvector mesonsor bypions that were misiden-tified as kaons. Note that if these assumptions should not hold, thecorrected RK values wouldbefurther decreasedwithrespect

tothe results presentedin thisLetter, i.e. the disagreement with pQCDexpectationswouldbeevenstronger.

InFig. 5,the dependence of RK onthe virtual-photonenergy

ν

in bins of the reconstructed z variable is shown for the first x-bin. Aclear

ν

-dependenceof RK isobserved forall z-bins,

ex-ceptthelastone.Withinexperimentaluncertainties,theobserved dependence on

ν

is linear and in the last bin a constant. Note that atmost15% of theobserved variation of RK with

ν

can be

explainedby the fact that in a given z-bin events atdifferent

ν

havesomewhatdifferentvaluesofx and Q2_{.Theobservedstrong}

ν

dependencesuggeststhatforlargervalues of

ν

theratio RK is

closertothe lower limitexpected frompQCD thanit isthe case forsmallervaluesof

ν

.Numericalvaluesforthe

ν

dependenceof RK inbinsofzrecaregivenforbothx-binsinRef. [27].

Inthisanalysis,thelargestdiscrepancybetweenpQCD expecta-tionsandexperimentalresultsisobservedintheregionoflargez andsmall y,i.e. small

ν

. Asexactly inthisregion thepreviously published COMPASS data [3] hadshownthe largest tensionwith the NLO pQCD fits ofFFs, see Section 1, the presentresults pro-videadditionalevidencethatthistensionisofphysicalorigin.

The observed violation of the pQCD expectations for the

charged-kaon multiplicity ratio at large values of z may be in-terpretedasfollows.Iftheproducedkaoncarriesalargefractionz ofthevirtual-photonenergy,thereisonly asmallamountof en-ergy left to fulfil conservation laws as e.g. those for strangeness numberandbaryonnumber,whichare nottakenintoaccount in the pQCD expressions for the SIDIScross section. The larger the valueofz,thesmalleristhenumberofpossiblefinalstatesinthe process understudy.The naturalvariableto studythe “exclusivi-ty”ofaprocessisthemissingmass,whichisapproximatelygiven byMX

=



M2

p

+

2Mp

ν

(

1

−

z

)

−

Q2

(

1

−

z

)

2.Asthefactor

ν

(

1

−

z

)

appears inthemissingmassdefinition,boththe z andthe

ν

de-pendenceofRK maybedescribedsimultaneouslybythisvariable.

Fig.6showsthat RKasafunctionof MX followsarathersmooth

behaviour.ThedisagreementbetweenourdataandthepQCD pre-dictionssuggests thata correction within thepQCD formalismis neededinordertotakeintoaccountthephasespaceavailablefor thehadronisationofthetargetremnant.Weobservethatourdata can bereconciled withthepQCD NLO prediction(RK larger than

about0.4)onlyabovetheratherhighMX valueofabout4GeV/c2, whichis rathersurprising(see e.g. Ref. [28]).Since thedominant term in MX is

∝

√

ν

(

1

−

z

)

, this observation also suggests that forexperimentswithaccessible valuesof

ν

smallerthan thoseat COMPASS, thedisagreement withpQCD calculationsand possible deviations from these expectations may already be observed at smallervaluesofz.

Fig. 6. TheK−overK+multiplicityratiopresentedasafunctionofMX.Seetextfor

details. 7. Summary

Inthis Letter, the K− over K+ multiplicity ratio RK measured

indeep-inelastickaonleptoproductionatlargevaluesofz is pre-sented for the first time. It is observed that the RK values fall

belowthelower limitscalculated atLOandNLO accuracyinthe pQCDformalism. Inaddition,we observethat thekaon multiplic-ityratio RK strongly dependsonthe missingmassinthe

single-inclusive kaon production process. Altogether, our observations suggest that moretheory effort may be requiredin orderto un-derstandkaonproductionathighz.Inparticular,withinthepQCD formalismanadditionalcorrectionmayberequiredthattakesinto accountthephasespaceavailableforhadronisation.

Acknowledgements

WewouldliketothankD.Stamenovforusefuldiscussions.We gratefullyacknowledgethesupportoftheCERNmanagement and staffandtheskillandeffortofthetechniciansofourcollaborating institutes.Thisworkwasmadepossiblebythefinancialsupportof ourfundingagencies.

Appendix A. Procedureforz-unfolding

A typical unfolding procedure produces a covariance matrix with non-negligible off-diagonal matrix elements. These correla-tions areimportantandinmanycasescannot be neglected,asit isalsoemphasisedinRef. [29].Incertainphenomenological analy-sesofpublishedmultiplicitydata,however,theseimportantpieces ofinformationare erroneouslyneglected, whichmaylead to im-properdatatreatment andthus toincorrectconclusions.In order topreventsuchproblems,wechoseasimpleunfoldingmethodin ourmain analysis. We note that anycorrectlyperformed unfold-ing procedure can only decrease the value of RK measured at a

givenvalue ofzrec, sothat thechoiceof theunfoldingprocedure

cannotpossibly explainthediscrepancyobservedbetweenpQCD predictionsandCOMPASSresults.

As an exampleofa moresophisticated z-unfoldingmethod,a procedureispresentedthatassuresasmoothbehaviourofthe re-sultingcharged-kaonmultiplicityratio.BasedonMCdataa smear-ingmatrixiscreated,inwhichtheprobabilitiesarestoredthatthe kaonwitha generated value z that belongsto a certain zgen-bin

is reconstructed in a certain zrec-bin. The widthof the z-binsis

chosen to be 0.05 andvalues of zrec up to 1.10are studied.The

obtainedsmearingmatrixisgiveninRef. [27] assupplemental ma-terial.Inthenextstep,afunctionalformfortheK±multiplicities

Table A.1

The z-unfolded RK defined as zzminmax

dMK− dz dz/ zmax zmin dMK+ dz dz,

wherezmin(max)denotebinlimitsinz.Thedatabelow(above)

x=0.05 arepresentedinthetop(bottom)partofthetable. Bin zmin zmax RK± δRK,stat.± δRK,syst.

1 0.75 0.80 0.416±0.009±0.018 2 0.80 0.85 0.360±0.010±0.017 3 0.85 0.90 0.289±0.009±0.014 4 0.90 0.95 0.200±0.014±0.011 5 0.95 1.00 0.085±0.022±0.007 1 0.75 0.80 0.237±0.006±0.011 2 0.80 0.85 0.202±0.006±0.010 3 0.85 0.90 0.165±0.006±0.009 4 0.90 0.95 0.123±0.009±0.007 5 0.95 1.00 0.068±0.016±0.005 Table A.2

Thecorrelation matrixrelatedtototaluncertaintiesofthedata presentedinTableA.1.

Bin 1() ₂() ₃() ₄() ₅() 1 1.00 0.99 0.89 0.39 −0.18 2 0.99 1.00 0.94 0.47 −0.12 3 0.89 0.94 1.00 0.74 0.21 4 0.39 0.47 0.74 1.00 0.81 5 −0.18 −0.12 0.21 0.81 1.00 1 1.00 0.98 0.84 0.37 −0.15 2 0.98 1.00 0.93 0.50 −0.04 3 0.84 0.93 1.00 0.78 0.30 4 0.37 0.50 0.78 1.00 0.82 5 −0.15 −0.04 0.30 0.82 1.00

is assumedinthe‘true’ phase spacefordata,whichforMC data corresponds tothephase spaceofgeneratedvariables.Forthefit of therealdata,the functionalform

α

·

exp

(β

z

)(

1

−

z

)

γ isused. Thisfunctionisintegratedinbinsofzgen,whicharedefinedbythe

smearingmatrix.Inthisway,avectorofexpectationvaluesis ob-tained in the‘true’ phase space. Thisvector ismultiplied by the smearingmatrix,resultinginexpectationvaluesforkaonyieldsin the reconstructed phase space. The yield predictions obtained in this wayare directly compared withthe experimental valuesby calculatinga

χ

2 _{value.Thisvalueisminimisedtofindoptimal}

pa-rametersforthefittingfunction.Inordertoobtaintheuncertainty oftheunfoldedratio,thebootstrapmethodisusedwith400 repli-cas ofourdata[30].At agivenvalueof z,theuncertaintyofthe ratioistakenasRootMeanSquarefromthereplicasdistribution. The effectofunfoldingis rathersmallforall binsexceptthelast one.TheobtainedresultsaresummarisedinTableA.1andthe cor-relationmatrixisgiveninTableA.2.

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u

,

G. Sbrizzai

w

,

x

,

P. Schiavon

w

,

x

,

H. Schmieden

d

,

E. Seder

t

,

A. Selyunin

g

,

L. Silva

k

,

L. Sinha

f

,

S. Sirtl

h

,

M. Slunecka

g

,

F. Sozzi

x

,

J. Smolik

g

,

A. Srnka

e

,

D. Steffen

i

,

o

_,

_{M. Stolarski}

k

,

∗

_,

_{O. Subrt}

i

,

r

_,

_{M. Sulc}

j

_,

H. Suzuki

af

,

5

,

A. Szabelski

w

,

x

,

ac

,

T. Szameitat

h

,

13

,

P. Sznajder

ac

,

M. Tasevsky

g

,

S. Tessaro

x

,

F. Tessarotto

x

,

A. Thiel

c

,

J. Tomsa

q

,

F. Tosello

z

,

V. Tskhay

n

,

S. Uhl

o

,

B.I. Vasilishin

aa

,

A. Vauth

i

,

B.M. Veit

l

,

J. Veloso

a

,

A. Vidon

t

,

M. Virius

r

,

S. Wallner

o

,

M. Wilfert

l

,

R. Windmolders

d

,

K. Zaremba

ae

,

P. Zavada

g

,

M. Zavertyaev

n

,

E. Zemlyanichkina

g

,

18

,

M. Ziembicki

ae

,

a_{UniversityofAveiro,Dept.ofPhysics,3810-193Aveiro,Portugal}

b_{UniversitätBochum,InstitutfürExperimentalphysik,44780Bochum,Germany}14,15 c_{UniversitätBonn,Helmholtz-InstitutfürStrahlen- undKernphysik,53115Bonn,Germany}14 d_{UniversitätBonn,PhysikalischesInstitut,53115Bonn,Germany}14

e_{InstituteofScientificInstruments,ASCR,61264Brno,CzechRepublic}16

f_{MatrivaniInstituteofExperimentalResearch&Education,Calcutta-700030,India}17 g_{JointInstituteforNuclearResearch,141980Dubna,Moscowregion,Russia}18 h_{UniversitätFreiburg,PhysikalischesInstitut,79104Freiburg,Germany}14,15 i_{CERN,1211Geneva23,Switzerland}

j_{TechnicalUniversityinLiberec,46117Liberec,CzechRepublic}16 k_{LIP,1000-149Lisbon,Portugal}19

l_{UniversitätMainz,InstitutfürKernphysik,55099Mainz,Germany}14 m_{UniversityofMiyazaki,Miyazaki889-2192,Japan}20

n_{LebedevPhysicalInstitute,119991Moscow,Russia}

o_{TechnischeUniversitätMünchen,PhysikDept.,85748Garching,Germany}14,4 p_{NagoyaUniversity,464Nagoya,Japan}20

q_{CharlesUniversityinPrague,FacultyofMathematicsandPhysics,18000Prague,CzechRepublic}16 r_{CzechTechnicalUniversityinPrague,16636Prague,CzechRepublic}16

s_{StateScientificCenterInstituteforHighEnergyPhysicsofNationalResearchCenter‘KurchatovInstitute’,142281Protvino,Russia} t_{IRFU,CEA,UniversitéParis-Saclay,91191Gif-sur-Yvette,France}15

u_{AcademiaSinica,InstituteofPhysics,Taipei11529,Taiwan}21

v_{TelAvivUniversity,SchoolofPhysicsandAstronomy,69978TelAviv,Israel}22 w_{UniversityofTrieste,Dept.ofPhysics,34127Trieste,Italy}

x_{TriesteSectionofINFN,34127Trieste,Italy} y_{UniversityofTurin,Dept.ofPhysics,10125Turin,Italy} z_{TorinoSectionofINFN,10125Turin,Italy}

aa_{TomskPolytechnicUniversity,634050Tomsk,Russia}23

ab_{UniversityofIllinoisatUrbana-Champaign,Dept.ofPhysics,Urbana,IL61801-3080,USA}24 ac_{NationalCentreforNuclearResearch,00-681Warsaw,Poland}25

ad_{UniversityofWarsaw,FacultyofPhysics,02-093Warsaw,Poland}25

ae_{WarsawUniversityofTechnology,InstituteofRadioelectronics,00-665Warsaw,Poland}25 af_{YamagataUniversity,Yamagata 992-8510,Japan}20

*

Correspondingauthors.

E-mailaddresses:[email protected](O.Yu. Denisov),[email protected](J.M. Friedrich),[email protected](M. Stolarski).

† _Deceased.

1 _{AlsoatInstitutoSuperiorTécnico,UniversidadedeLisboa,Lisbon,Portugal.}

2 _{AlsoatDept.ofPhysics,PusanNationalUniversity,Busan609-735,RepublicofKoreaandatPhysicsDept.,BrookhavenNationalLaboratory,Upton,NY11973,USA.} 3 _{AlsoatAbdusSalamICTP,34151Trieste,Italy.}

4 _{SupportedbytheDFGclusterofexcellence‘OriginandStructureoftheUniverse’(www.universe-cluster.de)(Germany).} 5 _{AlsoatChubuUniversity,Kasugai,Aichi487-8501,Japan.}

6 _{AlsoatDept.ofPhysics,NationalCentralUniversity,300JhongdaRoad,Jhongli32001,Taiwan.} 7 _{AlsoatKEK,1-1Oho,Tsukuba,Ibaraki305-0801,Japan.}

8 _{AlsoatMoscowInstituteofPhysicsandTechnology,MoscowRegion,141700,Russia.} 9 _{Presentaddress:RWTHAachenUniversity,III.PhysikalischesInstitut,52056Aachen,Germany.} 10 _{AlsoatYerevanPhysicsInstitute,AlikhanianBr.Street,Yerevan,Armenia,0036.}

11 _{AlsoatDept.ofPhysics,NationalKaohsiungNormalUniversity,KaohsiungCounty824,Taiwan.} 12 _{AlsoatUniversityofEasternPiedmont,15100Alessandria,Italy.}

13 _{SupportedbytheDFGResearchTrainingGroupProgrammes1102and2044(Germany).} 14 _{SupportedbyBMBF–BundesministeriumfürBildungundForschung(Germany).} 15 _{SupportedbyFP7,HadronPhysics3,Grant283286(EuropeanUnion).}

16 _{SupportedbyMEYS,GrantLG13031(CzechRepublic).} 17 _{SupportedbyB.Senfund(India).}

18 _{SupportedbyCERN-RFBRGrant12-02-91500.}

19 _{SupportedbyFCT–FundaçãoparaaCiênciaeTecnologia,COMPETEandQREN,GrantsCERN/FP116376/2010,123600/2011andCERN/FIS-NUC/0017/2015(Portugal).} 20 _{SupportedbyMEXTandJSPS,Grants18002006,20540299,18540281and26247032,theDaikoandYamadaFoundations(Japan).}

21 _{SupportedbytheMinistryofScienceandTechnology(Taiwan).} 22 _{SupportedbytheIsraelAcademyofSciencesandHumanities(Israel).}

23 _{SupportedbytheRussianFederationprogram“Nauka”(ContractNo.0.1764.GZB.2017)(Russia).} 24 _{SupportedbytheNationalScienceFoundation,Grantno.PHY-1506416(USA).}