Testing the universality of the Collins function in pion-jet production at RHIC

Contents lists available atScienceDirect

Physics

Letters

B

www.elsevier.com/locate/physletb

Testing

the

universality

of

the

Collins

function

in

pion-jet

production

at

RHIC

Umberto D’Alesio

a,b,

∗

,

Francesco Murgia

b

,

Cristian Pisano

c

a_{DipartimentodiFisica,UniversitàdiCagliari,CittadellaUniversitaria,I-09042Monserrato(CA),Italy} b_{INFN,SezionediCagliari,CittadellaUniversitaria,I-09042Monserrato(CA),Italy}

c_{DipartimentodiFisica,UniversitàdiPavia,&INFN,SezionediPavia,ViaA.Bassi6,I-27100,Pavia,Italy}

a

r

t

i

c

l

e

i

n

f

o

a

b

s

t

r

a

c

t

Articlehistory: Received10July2017

Receivedinrevisedform14August2017 Accepted18August2017

Availableonline25August2017 Editor:A.Ringwald

Keywords: QCDphenomenology Polarizationeffects

Transversemomentumdependent distributions

Universality

By adoptingageneralisedpartonmodel approachatleading orderinQCD, includingspinandintrinsic parton motion effects, we study the Collins azimuthal asymmetries for pions within a large-pT jet produced at mid-rapidity in polarised hadronic collisions. Using available information on the quark transversity distributions and the pion Collins functions, as extracted from semi-inclusive deeply inelastic scattering and e+e−→h1h2X processes, wecompute estimatesforthe Collinsasymmetries

in kinematical configurations presently investigated at RHIC by the STAR Collaboration. Collins-like asymmetries,involvinglinearlypolarisedgluons,arealsoconsidered.Ourpredictions,comparedagainst available preliminary data, show a very good agreement, even if some discrepancies, to be further scrutinized boththeoretically andexperimentally, appearinthetransverse momentumdependenceof the Collinsasymmetry.Theseresultsareinfavour ofthepredicted universalityoftheCollinsfunction andofamild,ifany,evolutionwiththehardscaleoftheasymmetries.

©2017TheAuthor(s).PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense (http://creativecommons.org/licenses/by/4.0/).FundedbySCOAP3.

1. Introduction

Transversesingle-spin asymmetries(SSAs) are a long-standing challengeandastrikingissueforcollinearleading-twist perturba-tiveQCD.Well-knownexamplesare:thelargetransverse polarisa-tionof



hyperonsproducedinunpolarised pp,p A collisions;the sizablepionSSAsmeasuredinpolarisedpp collisions,firstatfixed targetexperimentsandrecentlyatRHIC,atmuchlargerc.m. ener-giesandpiontransverse momentum; theazimuthal asymmetries measuredinsemi-inclusivedeeplyinelasticscattering(SIDIS) and ine+e− collisions.Theseobservablesandtheir understandingare indeedcrucialfora fullknowledgeofthenucleon structure, con-cerning in particular parton orbital motion andangular momen-tum,andtheconsequencesforspinsumrulesandtheviolationof helicityselectionrulesinhigh-energyexclusiveprocesses.Itisby nowclearthatthespinoftheprotoncannotbeexplainedsimply by the sumofthe spinsof its constituents,including seaquarks and gluons, and that a crucial role could be played by parton orbital angularmomentum. The so-called

transverse-momentum-*

Correspondingauthorat:DipartimentodiFisica,UniversitàdiCagliari,Cittadella Universitaria,I-09042Monserrato(CA),Italy.

E-mailaddresses:umberto.dalesio@ca.infn.it(U. D’Alesio),

francesco.murgia@ca.infn.it(F. Murgia),cristian.pisano@pv.infn.it(C. Pisano).

dependent (TMD) approach, including spin and intrinsic parton motion,isbynowaccreditedasoneofthetheoreticalformalisms able toaccount formanyofthesespin effects,atleastincertain kinematicalconfigurations(seee.g. Refs.[1–3] forgeneralreviews onSSAsandTMDsandRefs. [4–6]forthelatesttheoretical devel-opments).

For an alternative approach, extending the usual collinear scheme with inclusion of higher-twist effects and gluon–quark– gluon correlators, see for example Refs. [7,8] and references therein.

IntheTMD approachthespinasymmetriesare ultimatelydue to TMD partondistribution (TMD-PDF)and fragmentation (TMD-FF) functions. The most relevant ones are the Sivers distribu-tion function[9]andthe Collinsfragmentationfunction[10].The leading-twist TMD Collins FFdescribesthe asymmetry inthe az-imuthaldistributionofanunpolarisedhadronaroundthedirection ofmotionofthetransverselypolarisedfragmentingparentquark. ItischiraloddandnaivelyT-odd,andisnonvanishingonlyifthe transverse motionoftheproduced hadron(w.r.t. the quark direc-tionofmotion)isexplicitlytakenintoaccount.

AccordingtoTMDfactorisationtheorems,validfortwo energy-scale processes like SIDIS, e+e− annihilation and Drell–Yan pro-cesses,theCollinsFFisexpectedtobeuniversalandprocess inde-pendent[11].Itisresponsiblefortheazimuthalcorrelationinthe http://dx.doi.org/10.1016/j.physletb.2017.08.023

0370-2693/©2017TheAuthor(s).PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense(http://creativecommons.org/licenses/by/4.0/).Fundedby SCOAP3_.

distributionoftwohadronsproducedinoppositejetsine+e− col-lisionsandforaspecificazimuthal modulationmeasured inSIDIS withatransverselypolarisedtarget.Ininclusivesingle-hadron pro-ductioninpolarised pp collisionsthe situationisdefinitely more involved,both theoreticallyandphenomenologically, since,within ageneralisedpartonmodel(GPM)approach(apartonmodelwith inclusionofspinandtransversemomentumeffects),theSiversand Collinseffectscannotbedisentangled(seeRefs.[12,13]).Moreover, aproofoffactorisationintermsofTMDsforsuchprocesses(where onlyone energyscale is present) isstill lacking [14] and poten-tialfactorisation breaking effects could be expected [15]. Testing ofuniversalityisthereforeaverynon-trivialtopic,sensitiveto im-portantaspectsofQCD.

Onthe other hand,thestudy oftheazimuthal distributionof leading hadrons produced in the fragmentation of a large trans-verse momentum jet offers a unique opportunity for studying TMDsaswellasfactorisationbreakingeffectsinhadroniccollisions and,by comparisonwithSIDISande+e− annihilations,get infor-mationon theprocess dependenceofthe TMDfunctions. For in-stance,adetailedstudyoftheSiversfunctioninthesameprocess hasbeenpresentedinRef. [16].Inthisrespecttheseobservables representa verypowerfultestinggroundofourunderstandingof SSAsandTMDeffects.

TheCollins asymmetryforthe p↑p

→

jet

π

X processatlarge rapidity was first considered in Ref. [17], accounting for parton transversemomentaonlyinthefragmentationprocessandgiving atthesametime theproofoffactorisation(atleastatthelowest order).TheapproachwasgeneralisedinRef.[18]includingparton motionalsointheinitialdistributions,withinaTMD phenomeno-logical model, and providing the full leading-twist structure for azimuthalasymmetries.A detailedanalysisforseveralinteresting asymmetriesinkinematicalconfigurations reachable atRHICwas alsopresented,whilesubsequently,inRef. [19],ashortdedicated reviewandfurther phenomenologicalstudies were collected. No-tice that, while the complete k_⊥ treatment adopted in the GPM approachassumesfactorisationandremains tobechecked exper-imentally,itallowsamuchricherphenomenologicalanalysis,also inviewoftestingpotentialfactorisationbreakingeffects.Itis nev-ertheless true that for the Collins asymmetry this model would give results similar to those obtainedadopting the collinear ap-proachforthejetproductionmechanismofRef.[17].

Sincethen,thefirstpreliminarydatafortheCollinsasymmetry inthemid-rapidityregionhavebeenreleasedbytheSTAR Collab-oration,bothat

√

s

=

200and500GeV[20].

Together with data fromSIDIS and e+e− collisions, these re-sultsallowforthefirsttimeadirecttestoftheuniversalityofthe Collinsfunction (aswell asof transversity)andtheeffective role ofitsscaledependence.

Therefore, in thisletter we apply the TMD GPM approach to the Collins asymmetry in pion-jet production (as presented in Ref. [18]), keepingfixed the parameterizations of the transversity andCollinsfunctionsasobtainedbyfittingtheSIDISande+e− re-sults.Byimposing thesamekinematicalcutsasadoptedatRHIC, we give predictions for the Collins asymmetry in p↑p

→

jet

π

X

processesandcompareourresultswiththenewpreliminarySTAR data,lookingforinformationonuniversalityandscaledependence of the Collins function. For completeness, we will also consider the gluon Collins-like asymmetry, due to the convolution of the TMDdistributionoflinearlypolarisedgluonsinsideatransversely polarisedprotonwiththeTMDfragmentationfunction oflinearly polarised gluons into an unpolarised hadron. In such a case we willshowhowpresentpreliminarydata[21,22]couldhelpin con-straining the product of these two completely unknown TMDs. A separate and more direct extraction of the linearly polarized gluondistributionwouldbepossiblebylookingatdijetandheavy–

quark pair production in electron–proton collisions at a future ElectronIonCollider[23].

Thepaperisorganizedasfollows:insection2werecallthe ba-sicideas ofthe formalism,skipping all detailsof thecalculations that canbe foundinRef. [18]; insection3 wepresentour theo-reticalestimatesandcomparethemwiththeavailablepreliminary experimentalresults;conclusions andopen issuesaregatheredin section4.

2. Formalism

The azimuthal moment of the Collins asymmetry for the

p↑p

→

jet

π

X processisdefinedasfollows: Asin(φS−φπH) N

(

pj

,

z

,

k⊥π

)

=

2



d

φ

Sd

_

φ

πH sin

(φ

S

− φ

πH

)

[

d

σ

(φ

s

, φ

πH

)

−

d

σ

(φ

s

+

π

, φ

πH

)

]

d

φ

Sd

φ

πH

[

d

σ

(φ

s

, φ

πH

)

+

d

σ

(φ

s

+

π

, φ

πH

)

]

,

(1)

whered

σ

(φS

,

φ

_πH

)

isashorthandnotationfortheinvariant differ-entialcrosssection

Ejd

σ

p(S,φS)p→jetπ(φ H π)X d3_p jdz k⊥πdk⊥πd

φ

πHd

φ

S

,

(2)

andthenumeratorinEq.(1),takingintoaccountthatanother al-lowedtermbecomes negligibleuponintegrationoverthepartonic azimuthalphases,isschematicallygivenas

N

[

Asin(φS−φπH) N

] ∼

h q 1

(

xa

,

k 2 ⊥a

)

⊗

f1

(

xb

,

k2_⊥b

)

⊗  ˆ

σ

⊗

H⊥ q 1

(

z

,

k 2 ⊥π

) .

(3)

Notice that inthe denominator of theSSA, that is twice the un-polarisedcrosssection,we includeallkindofpartons,quarksand gluons.

For all details we refer to Ref. [18]. Here we only recall that

p_j is the jet three-momentum, that will be expressed in terms of its pseudorapidity

η

j and transverse momentum pjT; z is the fractionofthejet momentumcarriedby theobservedpion; k_⊥π

isthetransverse momentumofthepionwithrespecttothe par-entparton(the jetinourleading-order approach),and

φ

_{π is}H the azimuthal angleofthe pionmomentum,measured inthejet he-licity frame; S isthe(transverse)polarisationvectoroftheinitial protonbeam, forming an angle

φS

withthe jet productionplane (the x-z plane) in the pp c.m. reference frame. The azimuthal factor sin

(φS

− φ

_πH

)

in the numerator of Eq. (1) singles out the Collinscontribution,giveninEq.(3)astheconvolutionoftheTMD transversitydistributionforquarksinsidethepolarisedproton,hq₁, withtheTMDunpolariseddistributionforthepartons(quarksand gluons) inside theunpolarised proton, f1,the Collins function in the fragmentation process, H⊥₁q, and the partonic spin transfer,



σ

ˆ

.

If, alternatively, one uses as azimuthal weight the factor sin

(φS

−

2

φ

_πH

)

then the so-called Collins-like contribution is sin-gled out, as discussed briefly in the sequel. This term is related to the convolution of the TMD distribution for linearly polarised gluons in the polarised proton (which vanishes in the collinear configuration) withthe unpolarised TMD distribution in the un-polarisedprotonandtheCollins-likegluonFFinthejet fragmen-tation process (that parameterises the fragmentation of linearly polarised gluons into an unpolarised hadron). Other possible az-imuthalasymmetries,liketheSiverseffect,willnotbeconsidered inthispaper[18].

Inthecalculations,theTMDPDFsandFFsareparameterised us-ing asimple form, wherethelight-cone momentum fractionand

Table 1

KinematicalcutsadoptedbytheSTARCollaborationandusedinthepresenttheoreticalestimates.Thevaluesinroundbracketsfor

√

s=500 GeVrefertothecutsfortheCollins-likeasymmetries.

Kinematical cut √s=200 GeV √s=500 GeV

Jet cone radius R=0.6 R=0.5

Min(pjT) 10.0 GeV 22.7 (6.0) GeV

Max(pjT) 31.6 GeV 55.0 (13.8) GeV

Jet pseudorapidity |ηj| ≤1 |ηj| ≤1

Pion pTminimum 0.2 GeV 0.2 GeV

Pion pTmaximum 30.0 GeV 30.0 GeV

PionR=(φπ− φj)2+ (ηπ−ηj)2 R>0.1 R>0.04

Pion z= |pπ|/|pj| 0.1<z<0.6 0.1<z<0.8

Pion jT≡k⊥π 0.125<k⊥π<4.5 GeV 0.1<k⊥π<2.0 GeV

transverse momentum dependences are factorised, and only the DGLAPQCD evolution ofthecollinear parts,choosing pjT as fac-torisation scale,is taken intoaccount. The main reasonis that a proper TMD evolution framework for such a process is still not available and could be potentially differentfrom what is usually adopted inprocesses like SIDIS. Moreover, it is interesting by it-selftosee towhatextenttheordinary collinearevolutionisable todescribetheavailable data,which,eveniftakenatvery differ-entpjT values,donotshowanysignificantscaledependence[20]. Furtherstudyinthisdirectionwouldbecertainlyhelpful.

Thetransverse-momentumdependentpartistakentobe Gaus-sian (times proper k_⊥ factors) and flavour independent. All free parameters entering the parametrisation of the transversity dis-tribution andof the Collins functionsare fixed by fitting experi-mentaldatafortheCollinsasymmetryinSIDISpionproductionby theHERMESandCompassCollaborations,andfortwo-hadron az-imuthalcorrelationsin e+e−

→

π π

X processesbythe Belleand BaBarCollaborations.

Noadditionalfreeparameterisintroducedinouranalysis,and thetheoreticalcurves,comparedwiththepreliminarySTAR exper-imental data on the Collins asymmetry in p↑p

→

jet

π

X , are to be considered asdirectpredictions basedon aTMD factorisation scheme.

In the following we will adopt different sets of the quark transversity and Collins functions, which we will refer to asthe SIDIS 1[24],theSIDIS 2[25] setsandthe2013fit[26].The rea-son of thischoice is that the SIDIS 1 andSIDIS 2 sets are those used inour first study [18]: in thisrespect, by keeping them in the present analysis we emphasise its aspect of being a natural extensionofourformerwork,withpredictionssomehowobtained directlyfromthere.Whiletheyarewellrepresentativeofthe avail-ableextractionsoftheseTMDsandtheiruncertainties,forthesake ofcompletenessandtokeepourstudyup-to-datewehave consid-eredthemorerecent2013fit aswell. Insuch a casein showing ourestimateswewillalsoprovidethecorrespondingstatistical un-certainty bands, calculated following the procedure described in AppendixAofRef.[27].

Acommentontheuse,andimpact,ofthemostrecent extrac-tion[28]ofthetransversitydistributionandtheCollinsFFwillbe givenbelow.

Thesesets,besidessome differencesintheinitial assumptions andinthedatausedfortheirextraction,differinthechoiceofthe collinearfragmentationfunctions.Moreprecisely,fortheSIDIS 1fit theKretzerFFset[29]wasadopted,whilefortheSIDIS 2andthe 2013fitstheDSSFFset[30]wasemployed.Thelatter,in particu-lar,incorporatesamoresizeablegluonFFw.r.t. thecorresponding oneintheKretzerset,thatcouldplayaroleintheprocessunder consideration.

Noticethatinallcasestheextractionofthequarktransversity distributionisconstrainedonlyuptox



0

.

3.

3. Results

The STAR Collaboration at RHIC has collected preliminary re-sults on the Collins andCollins-like asymmetries for the p↑p

→

jet

π

X process in the midrapidity range (

|η

j

|

≤

1) at c.m. ener-gies

√

s

=

200 and500GeV[20–22].Table 1summarisesthemain kinematicalcutsrelevantforourphenomenologicalstudy.Wehave implemented all of them in our calculations. Notice that in our leading-order TMD approach the jet is identified with the final fragmenting partoncoming fromthe hard subprocesses andthat the STARCollaborationadoptstheanti-kt jetreconstruction algo-rithm. Fordetailedstudies ofthetransversemomentum distribu-tionofhadronswithinajetseeRefs. [31,32].Wealsoremarkthat some oftheabovecutsinducea cos

φ

_{π azimuthal}H dependencein the(un)polarisedcrosssections.Wehavecheckedthatthis depen-dencecancelsoutintheasymmetries.

In Fig. 1 we present our theoretical estimates of the Collins asymmetry for

π

± productionat

√

s

=

200 GeV (left panel)and 500 GeV(rightpanel)asafunctionofz,comparedwithSTAR pre-liminarydata[20],integratedovertheotherkinematicalvariables andimposingtheexperimentalcutsofTable 1.Ourpredictionsare givenforthesetsofthetransversitydistributionandtheCollinsFF discussedabove;inparticular,forthe2013fitwealsoshowtheir statistical uncertaintybands. The agreement betweentheory and dataisdefinitelyverygood.

Notice that, both at 200 and 500 GeV energies, the average momentum fractionofthequarks insidethepolarised proton(as well as inside the unpolarised one) is around 0.2. That means we areprobingthevalenceregionofthetransversity distribution and, moreimportantly,theregion whereitiswell constrainedby SIDIS data. Almost no energy dependence appears (see also the comments belowonthe exploredkinematicalregion). The differ-ences between the estimates obtained adopting the SIDIS 1 set w.r.t. thoseobtainedwiththeSIDIS 2set andthe2013fitare al-mostcompletelyduetothefactthattheunpolarisedcrosssection computedwiththeDSSFFset(associatedtotheSIDIS 2fitandthe 2013fit,andentering thedenominatoroftheSSA)isbigger than the onecomputedwiththeKretzerset,inthemedium andlarge

z region. This, in turn, can be traced back to the large leading-ordergluonFFofthisset,essentiallydrivenbyitsextractionfrom a globalfit including also pp

→

π

X data. We have to note that thecorrespondingnext-to-leading-orderextraction,notusedhere, wouldbemorestable.

InFig. 2wepresentourpredictionsfortheCollins asymmetry for

π

±productionasafunctionofpjT at

√

s

=

200 GeVcompared withSTARpreliminarydata.Herewenoticeadifferentbehaviour, w.r.t. what discussed above: at large pjT

≥

20 GeV, in fact, the SIDIS 2setgivesasymmetrieslargerthanthoseobtainedwiththe SIDIS 1set andthe 2013fit.The reasonistwofold:sincethe av-erage value of xa in thepolarized protonincreaseswith pjT (for instance at pjT



20 GeV

xa





0

.

3), fromone side the

transver-Fig. 1. TheCollinsasymmetryAsin(φS−φHπ)

N fortheprocessp↑p→jetπ±X ,asafunctionofz,atc.m. energy

√

s=200 GeV(leftpanel)and500GeV(rightpanel),compared withSTARpreliminarydata[20].Estimatesareobtainedbyintegratingovertheothervariables, imposingtheexperimentalcutsassummarisedinthelegend(seealso Table 1)andadoptingthreedifferentparameterisations:the2013fit(redsolidlines),withtheirstatisticaluncertaintybands,theSIDIS 1(greendashedlines)andSIDIS 2 (bluedottedlines)parameterisations.

Fig. 2. TheCollinsasymmetryAsin(φS−φπH)

N fortheprocessp↑p→jetπ±X ,asafunctionofpjT,atc.m. energy√s=200 GeV,separatelyforforward(leftpanel)andbackward

(rightpanel)rapidities,comparedwithSTARpreliminarydata[20].Estimatesareobtainedasinthepreviousfigure.

sityfunction isprobed inthe large-x regionwhere itis basically unconstrained.In particular, in the SIDIS 1 set aswell asin the 2013fitit resultsaccidentally more suppressedthan forSIDIS 2. From the other side, the role of gluons, both in the distribution andfragmentationfunctions, becomes lessrelevant, reducing the differencesbetweentheFFsets.

In Fig. 3, left panel, we show our estimates for the Collins asymmetryfor

π

± productionasafunctionoftheintrinsic trans-versemomentumofthepionw.r.t. thejetdirectioncomparedwith theSTAR preliminarydata, still adoptingthe TMD sets discussed above.Forcompleteness(seealsothediscussionbelow),asan ex-ample,intherightpanel weshow thecorrespondingunpolarised crosssectionfor

π

+ productionusingtheDSSFFset.

Wenoticethatatrelativelylargek_⊥π ourestimatesforthe

sin-gle spin asymmetry are in fair agreement with the data, while at lower values some discrepancies appear. More precisely, tak-ing intoaccount the statusofthe data,still preliminary, andthe factthatonewouldexpectabetteragreementinthelowk_⊥π

re-gion,amongthe possiblesourcesofthesediscrepancieswe could mention: i

)

a narrower k_⊥π dependence of the Collins functions

(w.r.t. theonesadoptedhere), withorwithouta narrower Gaus-sian width forthe unpolarised TMD FFs; ii

)

a different k_⊥π

be-haviour forfavoured and disfavoured TMD FFs; iii

)

an explicit z

dependencein the widths. Thesechanges indeed could shift the maximumoftheasymmetrytosmallerk_⊥π values,differentiating

alsoits behaviour for

π

+ and

π

− production,with a slight

im-pactonthek_⊥π -integratedpredictionsshownintheprevious

fig-ures.Ontheotherhand,thesamechangesshouldalsobechecked againsttheSIDISande+e− data,fromwhichtheCollinsFFs have been extracted. This is certainly an importantstudy that, imply-ing the use of an increasing number of free parameters, could be performed only in future extractions of TMDs. For the time being, it is worth reminding that we still do not have at our disposal precise dataon the k_⊥ dependenciesof theunpolarised crosssectionsandtheazimuthal asymmetriesin e+e− processes, and therefore an accurate knowledge of the explicit transverse momentum dependence of the unpolarised and Collins TMD FFs is still missing. For this reason data on unpolarised cross sec-tionsfortheprocessunderstudyhereandtheircomparisonwith the estimates shown in the right panel of Fig. 3 could be very useful.

One has also to recall that available information on this de-pendence is based on the analysis of SIDIS processes (namely multiplicitiesandazimuthal asymmetries),whereitappearstobe stronglycorrelatedwiththecorrespondingtransversemomentum dependenceintheTMDpartondistributionfunctions(see,for in-stance,Ref. [33]). Apreliminarystudyoftheimpactofthe Gaus-sianwidthsintheanalysisofSSAsisinprogress[34]and,dueto itsrelevance inthecontextoftheuniversalityissue,moreefforts, both from the experimental and the phenomenological point of view,aredefinitelynecessary.Forthesereasonstheresultsshown inFig. 3deservefurtherattention.

Fig. 3. TheCollinsasymmetryAsin(φS−φπH)

N fortheprocessp↑p→jetπ±X (leftpanel),andtheunpolarisedcrosssectionforπ+production(rightpanel),asafunctionofk⊥π (theintrinsictransversemomentuminthefragmentationprocess,alsodenotedasjT),atc.m. energy√s=200 GeV.PreliminarydataarefromtheSTARCollaboration[20]. Estimatesfortheasymmetryareobtainedasinthepreviousfigures,whiletheDSSFFsetisusedforthecalculationoftheunpolarisedcrosssection.Insuchacaseweshow theuncertaintybandobtainedvaryingthefactorisationscalebetweenpjT/2 and2pjT.

Somemoregeneralcommentsareinorder:

•

In our calculations the average intrinsic transverse momen-tumintheFF,

k_⊥π



,isintherange0.4–0.5GeVat200 GeV

and0.3–0.8 GeVat500 GeV;moreover,theaveragetransverse momentum of the jet,

pjT



, is around 12 GeV at 200 GeV and25–27 GeVat500 GeV.Thismeansweare intheproper regime to apply the TMD approach and that one should be sensibletoscaleevolutioneffects.Noticethattheangularcuts, characterised by the minimum distance of the charged pion fromthejetthrustaxis,havebeenchosentosamplethesame

xT values(xT

=

2pjT

/

√

s).

•

From thisanalysiswe can concludethat the Collins function manifests its universality not only in SIDIS and e+e− pro-cesses,butalsoin theazimuthal distributionofpions within jetsinhadroniccollisions. No compelling factorisation break-ingeffectsemerge.

•

The fact that the ordinary collinear evolution with the hard scale,asadoptedintheextractionoftheCollinsfunctionsand in thecurrent predictions, is ableto describe fairly well the experimental data atvery different energy scales suggests a verysmallroleofthemorecomplexTMDevolution.Thiscould berelatedtoapartialcancellationofitseffectsinaSSA,that isaratioofcrosssections.

•

The apparent tension between the poor description of the very low k_⊥π behaviour of the asymmetry for

π

+

produc-tion(Fig. 3) andtheoverallgoodagreementwiththedataas a function of z and pjT can be ascribed to the fact that the

k_⊥π -integratedobservablesarelesssensitivetothetransverse

momentumdependenceoftheCollinsfunctions.

•

Concerningtheuseofthelatestextraction[28]ofthe transver-sity distributionand the Collins FF, within the samescheme (thatisincludingonlytheDGLAPscaleevolution),thiswould givesmaller resultsforthe Collinsasymmetry.Thisismainly dueto the lower value of the Gaussian widthfor the TMD-FFs adopted in that analysis. This is a delicate issue and a detailed study of the impact of the Gaussian widths in the phenomenologicalanalysisofthe SiversandCollins effectsis underway[34].

•

Anothersetofdata[20](lowerplotoftheirFig. 4),exploringa muchlargerpionintrinsictransversemomentumw.r.t. thejet axis,isalmost compatiblewithzero andconsistentwithour estimates(notshownhere).

Before concludingthissectionwe wouldliketo comment fur-ther on the impact of the lack of knowledge of the transversity distribution atlarge x.This,forinstance,could berelevant in or-dertoproperlyassesspotentialfactorisationbreakingand/orTMD evolution effects.To this end in Fig. 4 we show some estimates obtained by sampling the parameter controlling the large-x be-haviour of thetransversity distribution, asdescribed in Ref. [12]. Notice that in such a case the shaded areas (referred to asscan bands inRef.[12])representonlytheenvelopeofallpossible val-ues of the asymmetry obtained following the above procedure. They are smaller than the corresponding statistical uncertainty bands shown in the left panels of Figs. 1 and 2, showing that such bands are not only dueto the uncertainty onthe transver-sity distribution atlarge x.Without entering into further details, andtakingintoaccounttheaboveconsiderations,theseresults al-low ustoconfirm onceagainourconclusions onthe universality of theCollins function andon thealmost negligible role ofTMD evolutioneffects.

3.1. Collins-likeazimuthalasymmetry

Followingthe generalresultspresentedinRef. [18],that show howto correlatedifferentangularmodulationstodifferentTMDs, STARhasextractedseveralotherangularmodulations[21,22].One example is the Collins-like asymmetry Asin(φS−2φπH)

U T as a function

of z,integratedoverthefullacceptanceandseparatedinforward and backwardscattering relative tothe polarised beam. InFig. 5

we presentthese STARpreliminary data,almost compatiblewith zero,togetherwithourmaximisedestimatesobtainedby saturat-ing thepositivity bounds1 ofthe two unknown TMDsrelated to the linearpolarisationof gluons(insidethepolarised protonand inthefragmentationprocess).Itisworthnoticingthatforthisdata settheaveragevalueofthemomentumfractioninsidetheprotons isaround0.05and

pjT



=

7–8 GeV.

As one can see, the maximised prediction suggests a possi-ble upperlimit of

∼

2% (independently ofthe FFset), while the present data fall well below this maximum, with the best pre-cision at lower values of z. Thus, these data represent the first phenomenological constraint on model predictions utilizing lin-early polarised gluons, beyond their positivity bounds. We can

1 _{Wemaximisethefirstk}

⊥-momentofthetransversemomentumdependent fac-toranduseforthex andz dependentpartsthecorrespondingunpolarisedgluon PDFandFF.

Fig. 4. TheCollinsasymmetryAsin(φS−φπH)

N fortheprocessp↑p→jetπ±X ,atc.m. energy

√

s=200 GeV,asafunctionofz (leftpanel)andpjT,(rightpanel),comparedwith

STARpreliminarydata[20].Theshadedareasrepresenttheenvelopeofallpossiblevaluesoftheasymmetry obtainedfollowingtheprocedureofRef.[12],fortwodifferent FFsets.

Fig. 5. TheCollins-likeasymmetry Asin(φS−2φπH)

N for theprocess p↑p→jetπ±X ,asa functionofz,atc.m. energy

√

s=200 GeV, separatelyfor forward(leftpanel) andbackward(rightpanel)rapidities,comparedwithSTARpreliminarydata[21,22].Estimatesarecomputedbyintegrating overtheothervariablesand imposingthe experimentalcutsassummarizedinthelegend(seealsoTable 1).BandsrepresentthemaximumvaluesinsizeoftheSSAobtainedbysaturatingthepositivityboundsof thetwogluonTMDsandadoptingtwodifferentFFsets.

safelysay that theseexperimental results implythat the product

of thesetwo unknown linearly polarised gluon TMDs cannot be largerthan 20–25%theproduct oftheir positivitybounds (focus-ingonthelowestz bin).

4. Conclusions

Inthis letter we havestudied the Collins and Collins-like az-imuthal asymmetries in the distribution of hadrons within jets producedinpolarisedproton–protoncollisions.ByapplyingaTMD approach atleading order, with inclusion of spin and transverse momentum effects,we have shownhow the preliminary dataof theCollinsasymmetrycollectedbytheSTARCollaborationatRHIC could be described in termsof a universal Collins function,with onlysomediscrepanciesinthedescriptionoftheirk_⊥π behaviour.

If confirmed by data, this would require a better understanding ofthe k_⊥ dependenciesof the unpolarised andthe Collins TMD FFs as extracted from current phenomenological analyses. These results, obtainedadopting available estimates of the transversity distributionandtheCollinsFF,asextractedfromindependentfits of the azimuthal asymmetries observed in SIDIS and e+e− pro-cesses,withstandardDGLAPQCDevolution,giveindeedanoverall good description of the STAR data in different kinematical con-figurations.No indication ofuniversality-breakingeffects emerges

and, quite interestingly,no compelling TMD evolution effects ap-pear. With some caution, this could be considered as the first phenomenologicalevidencefortheuniversalityoftheCollins func-tioninhadron–hadroncollisions.

TheCollins-likeasymmetry,involvinglinearlypolarisedgluons, bothinsidethepolarisedprotonandinthefragmentationprocess, hasalsobeenconsidered.InsuchacasetheSTARpreliminarydata allowforthefirstphenomenologicalconstraintonthesizeofthese TMDs.

Further study is certainly necessary, but these results open a newwindowinthefieldofTMDeffectsinhadroniccollisions.

Uponcompletionofthisanalysiswebecameawareofasimilar study,includingalsoTMDevolutioneffects,performedinRef.[35]. Theirresults,inagreementwithourfindings,supportour conclu-sions.

Acknowledgements

Wewouldliketo thankKevinAdkins,Jim Drachenberg,Renee FatemiandCarlGagliardiforusefulinformationontheSTAR pre-liminarydata.C.P. issupported bytheEuropeanResearchCouncil (ERC)underthe EuropeanUnion’s Horizon2020research and in-novationprogramme(grantagreementNo. 647981,3DSPIN).

References

[1] U.D’Alesio,F.Murgia,Azimuthaland singlespinasymmetriesinhard scat-teringprocesses,Prog.Part.Nucl.Phys.61(2008)394–454,http://dx.doi.org/ 10.1016/j.ppnp.2008.01.001,arXiv:0712.4328.

[2] V. Barone, F. Bradamante, A. Martin, Transverse-spin and transverse-momentumeffectsinhigh-energyprocesses,Prog.Part.Nucl.Phys.65(2010) 267–333,http://dx.doi.org/10.1016/j.ppnp.2010.07.003,arXiv:1011.0909. [3] E.C.Aschenauer,U.D’Alesio,F.Murgia,TMDsandSSAsinhadronicinteractions,

Eur. Phys. J. A 52 (2016) 156, http://dx.doi.org/10.1140/epja/i2016-16156-4, arXiv:1512.05379.

[4]J.Collins,FoundationsofPerturbativeQCD,CambridgeUniversityPress,2011. [5] M.G.Echevarria,A.Idilbi,I.Scimemi,FactorizationtheoremforDrell–Yanat

lowqT andtransversemomentumdistributionson-the-light-cone,J.High En-ergy Phys. 07 (2012) 002,http://dx.doi.org/10.1007/JHEP07(2012)002, arXiv: 1111.4996.

[6]J. Collins,T.C. Rogers,Connectingdifferent TMDfactorizationformalisms in QCD,arXiv:1705.07167.

[7] C.Kouvaris,J.-W.Qiu,W.Vogelsang,F.Yuan,Singletransverse-spin asymme-tryinhightransversemomentumpionproductioninpp collisions,Phys.Rev. D74(2006)114013,http://dx.doi.org/10.1103/PhysRevD.74.114013, arXiv:hep-ph/0609238.

[8] K.Kanazawa,Y.Koike,A.Metz,D.Pitonyak,M.Schlegel,Operatorconstraints fortwist-3functionsandLorentzinvariancepropertiesoftwist-3observables, Phys.Rev.D93(2016)054024,http://dx.doi.org/10.1103/PhysRevD.93.054024, arXiv:1512.07233.

[9] D.W.Sivers,Singlespinproductionasymmetriesfromthehardscatteringof point-likeconstituents, Phys.Rev.D41(1990)83,http://dx.doi.org/10.1103/ PhysRevD.41.83.

[10] J.C.Collins,Fragmentationoftransverselypolarized quarksprobed in trans-versemomentumdistributions,Nucl.Phys.B396(1993)161–182,http://dx. doi.org/10.1016/0550-3213(93)90262-N,arXiv:hep-ph/9208213.

[11] J.C. Collins, A. Metz, Universality of soft and collinear factors in hard-scatteringfactorization,Phys. Rev.Lett. 93(2004) 252001, http://dx.doi.org/ 10.1103/PhysRevLett.93.252001,arXiv:hep-ph/0408249.

[12] M. Anselmino, M. Boglione, U. D’Alesio, E. Leader, S. Melis, F. Murgia, A. Prokudin,RoleofCollinseffectinthesinglespinasymmetry AN inp↑p→

h X processes, Phys. Rev. D 86 (2012) 074032, http://dx.doi.org/10.1103/ PhysRevD.86.074032,arXiv:1207.6529.

[13] M.Anselmino,M.Boglione,U.D’Alesio,S.Melis,F.Murgia,A.Prokudin,Sivers effect and the single spin asymmetry AN in p↑p→h X processes, Phys. Rev.D88(2013)054023,http://dx.doi.org/10.1103/PhysRevD.88.054023,arXiv: 1304.7691.

[14] T.C. Rogers, Extra spin asymmetries from the breakdown of transverse-momentum-dependent factorization inhadron–hadron collisions, Phys. Rev. D 88 (2013) 014002, http://dx.doi.org/10.1103/PhysRevD.88.014002, arXiv: 1304.4251.

[15] J.Collins,J.W.Qiu,kTfactorizationisviolatedinproductionof high-transverse-momentum particles in hadron–hadron collisions, Phys. Rev. D 75 (2007) 114014,http://dx.doi.org/10.1103/PhysRevD.75.114014,arXiv:0705.2141. [16] U.D’Alesio,L.Gamberg,Z.-B.Kang,F.Murgia,C.Pisano,Testingtheprocess

de-pendenceoftheSiversfunctionviahadrondistributionsinsideajet,Phys.Lett. B704(2011)637–640,http://dx.doi.org/10.1016/j.physletb.2011.09.067,arXiv: 1108.0827.

[17] F. Yuan, Azimuthal asymmetric distribution of hadrons inside a jet at hadroncollider,Phys.Rev.Lett.100(2008)032003,http://dx.doi.org/10.1103/ PhysRevLett.100.032003,arXiv:0709.3272.

[18] U.D’Alesio,F.Murgia,C.Pisano,Azimuthalasymmetriesforhadron distribu-tionsinsideajetinhadroniccollisions,Phys.Rev.D83(2011)034021,http:// dx.doi.org/10.1103/PhysRevD.83.034021,arXiv:1011.2692.

[19] U.D’Alesio,F.Murgia,C.Pisano,CollinsandSiverseffectsinp↑p→jetπX : universalityand process dependence, Phys. Part. Nucl. 45 (2014) 676–691, http://dx.doi.org/10.1134/S1063779614040054,arXiv:1307.4880.

[20] J.K.Adkins,J.L.Drachenberg,Azimuthalsingle-spinasymmetriesofcharged pi-onsinjetsin√s=200GeVp↑p collisionsatSTAR,Int.J.Mod.Phys.Conf.Ser. 40(2016)1660040,http://dx.doi.org/10.1142/S2010194516600405.

[21] J.L.Drachenberg,ConstrainingTransversityandnucleontransverse-polarization structurethrough polarized-proton collisionsat STAR, in:Proceedings, 20th International Conferenceon Particles and Nuclei, PANIC14, Hamburg, Ger-many,August24–29,2014,pp. 181–184,http://inspirehep.net/record/1375601/ files/78.pdf.

[22] J.L.Drachenberg,Transversesingle-spinasymmetriesfrom p↑+p→jet+X andp↑+p→jet+π±+X at√s=500 GeV atRHIC,EPJWebConf.73(2014) 02009,http://dx.doi.org/10.1051/epjconf/20147302009.

[23] D.Boer,P.J.Mulders,C.Pisano,J.Zhou,Asymmetriesinheavyquarkpairand dijetproductionatanEIC,J.HighEnergyPhys.08(2016)001,http://dx.doi. org/10.1007/JHEP08(2016)001,arXiv:1605.07934.

[24] M.Anselmino,M.Boglione, U.D’Alesio,A.Kotzinian,F.Murgia,A.Prokudin, C. Türk,TransversityandCollinsfunctionsfromSIDISande+e−data,Phys.Rev. D75(2007)054032,http://dx.doi.org/10.1103/PhysRevD.75.054032, arXiv:hep-ph/0701006.

[25] M.Anselmino,M.Boglione, U.D’Alesio,A.Kotzinian,F.Murgia,A.Prokudin, S.Melis,Updateon transversityandCollinsfunctionsfromSIDISande+e− data, Nucl. Phys. Proc. Suppl.191 (2009)98–107, http://dx.doi.org/10.1016/ j.nuclphysbps.2009.03.117,arXiv:0812.4366.

[26] M.Anselmino,M.Boglione, U.D’Alesio,S.Melis,F.Murgia,A.Prokudin, Si-multaneousextractionoftransversityand Collinsfunctions fromnewSIDIS and e+e− data, Phys. Rev. D 87 (2013) 094019, http://dx.doi.org/10.1103/ PhysRevD.87.094019,arXiv:1303.3822.

[27] M. Anselmino, M. Boglione, U. D’Alesio, A. Kotzinian, S. Melis, F.Murgia, A. Prokudin,Siverseffectforpionandkaonproductioninsemi-inclusivedeep inelasticscattering,Eur.Phys.J.A39(2009)89–100,http://dx.doi.org/10.1140/ epja/i2008-10697-y,arXiv:0805.2677.

[28] M. Anselmino, M. Boglione, U.D’Alesio, J.O. Gonzalez Hernandez, S. Melis, F. Murgia, A. Prokudin, Collins functions for pions from SIDIS and new e+e− data:a firstglance at theirtransversemomentumdependence, Phys. Rev.D92(2015)114023,http://dx.doi.org/10.1103/PhysRevD.92.114023,arXiv: 1510.05389.

[29] S. Kretzer,Fragmentation functions from flavor inclusive and flavor tagged e+e− annihilations,Phys.Rev.D62(2000)054001,http://dx.doi.org/10.1103/ PhysRevD.62.054001,arXiv:hep-ph/0003177.

[30] D.deFlorian,R.Sassot,M.Stratmann,Globalanalysisoffragmentation func-tionsfor pionsand kaonsand their uncertainties, Phys. Rev.D 75 (2007) 114010,http://dx.doi.org/10.1103/PhysRevD.75.114010,arXiv:hep-ph/0703242. [31] T.Kaufmann,A.Mukherjee,W.Vogelsang,Hadronfragmentationinsidejetsin

hadroniccollisions,Phys.Rev.D92(2015)054015,http://dx.doi.org/10.1103/ PhysRevD.92.054015,arXiv:1506.01415.

[32]Z.-B.Kang,X.Liu,F.Ringer,H.Xing,Thetransversemomentumdistributionof hadronswithinjets,arXiv:1705.08443.

[33] A.Bacchetta,F.Delcarro,C.Pisano,M.Radici,A.Signori,Extractionofpartonic transversemomentumdistributionsfromsemi-inclusivedeep-inelastic scatter-ing,Drell–YanandZ-bosonproduction,J.HighEnergyPhys.1706(2017)081, http://dx.doi.org/10.1007/JHEP06(2017)081,arXiv:1703.10157.

[34] M.Anselmino,M.Boglione,U.D’Alesio,F.Murgia,A.Prokudin,inpreparation. [35]Z.-B.Kang,A.Prokudin,F.Ringer,F.Yuan,Collinsazimuthalasymmetriesof